Higher-Order Partial Differentiation Formulas for Extended Srivastava Hypergeometric Functions via a General Kernel-Based Beta Function

1. Introduction

The theory of special functions continues to play a fundamental role in modern mathematical analysis, particularly in the study of differential equations, integral transforms, and applied mathematical modeling. Classical hypergeometric functions, introduced in the works of Gauss and generalized by Euler, Appell, and Horn, have served as a cornerstone in this development due to their rich analytic structure and remarkable stability properties under various operators, (see, [1,2,3,4,5,6,7,8]).

In recent decades, there has been a growing interest in constructing extended and generalized classes of hypergeometric-type functions in order to describe increasingly complex systems arising in physics, engineering, and applied sciences. These extensions are often motivated by the need to incorporate additional parameters, coupling effects, and multivariable interactions that cannot be adequately captured by classical frameworks. In this direction, Srivastava-type multivariable hypergeometric functions have emerged as a powerful and flexible tool, unifying several known families of special functions within a single analytical structure, (see, [9,10,11,12,13]).

More recently, further extensions involving generalized beta-type kernels and parameter-dependent deformations have been introduced. These constructions not only enrich the algebraic structure of hypergeometric functions but also provide new avenues for analytical manipulation, particularly in the context of operational calculus and fractional analysis. Such generalized functions are typically defined through multi-index series expansions involving Pochhammer symbols, gamma functions, and extended beta-type operators, which allow for a highly structured parameter dependence, (see, [14,15,16,17,18,19,20,21,22,23,24,25]).

Motivated by these developments, we consider in this work the extended Srivastava-type hypergeometric functions $ᵏ{\widehat{H}}_A$, $ᵏ{\widehat{H}}_B$, and $ᵏ{\widehat{H}}_C$ defined by a general kernel-based beta function. These functions are represented by triple hypergeometric series involving coupled parameters and a generalized beta-function structure generated through an arbitrary kernel function. This kernel-based formulation provides additional flexibility and unifies various previously known extensions within a common analytical framework. In particular, the parameters $\kappa$, $\omega$, and $\tau$ play a central role in describing the interaction and deformation structure of the associated series representations.

A central motivation of this paper is to understand how these extended functions behave under repeated differentiation with respect to their deformation parameters. Unlike classical hypergeometric functions, where differentiation typically leads to simple parameter shifts, the presence of extended kernels and coupled Pochhammer structures leads to more intricate transformation rules. Nevertheless, as we demonstrate in this work, a remarkable invariance phenomenon still persists: the family of $ᵏ\widehat{H}$-functions remains closed under higher-order partial derivatives, up to explicit multiplicative Pochhammer factors and systematic parameter shifts.

From a theoretical perspective, this invariance property reveals a hidden algebraic stability of the extended Srivastava framework under differential operators. It also suggests that these functions form a natural class of objects for operational methods, particularly in problems involving repeated differentiation, convolution structures, and fractional-type generalizations. From an applied viewpoint, such properties are expected to be useful in the construction of exact solutions to generalized differential and integro-differential equations, especially those arising in mathematical physics and engineering models with memory or multi-scale effects.

The main contributions of this paper are as follows. First, we establish explicit r-th order partial differentiation formulas for $ᵏ{\widehat{H}}_A$, $ᵏ{\widehat{H}}_B$, and $ᵏ{\widehat{H}}_C$ with respect to $\kappa$, $\omega$, and $\tau$. Second, we show that these operations preserve the structural form of the underlying functions, leading to closed-form transformation rules. Third, we provide a unified treatment that highlights the symmetry and coherence among the three function classes within the extended Srivastava family.

The remainder of this paper is structured as follows. In Section 2, we present the preliminary definitions and classical results related to the gamma and beta functions, the Pochhammer symbol, and Srivastava’s triple hypergeometric functions. Section 3 introduces new extended Srivastava-type hypergeometric functions constructed via a general kernel-based beta function. In Section 4, we derive explicit higher-order partial differentiation formulas for the introduced function classes. Section 5 is devoted to the discussion and interpretation of the obtained analytical results, while concluding remarks and possible directions for future research are provided in Section 6.

2. Preliminaries

This section provides the fundamental mathematical concepts and auxiliary results required throughout the paper. We first recall several classical special functions, including the gamma and beta functions together with the Pochhammer symbol, which frequently arise in the theory of hypergeometric functions. Subsequently, we present the classical Srivastava triple hypergeometric functions $H_A$, $H_B$, and $H_C$, which serve as the basis for the generalized formulations investigated in this work.

Definition 1 

([26]). The gamma function for $\Re \left(\sigma \right)>0$ is defined by

$$\begin{array}{c}\hfill \Gamma \left(\sigma \right)={\int }_0^{\infty }{\omega }^{\sigma -1}exp\left(-\omega \right)d\omega .\end{array}$$

The gamma function satisfies the following properties:

• $\Gamma (\sigma +1)=\sigma \Gamma \left(\sigma \right)$,
• $\Gamma (\sigma +1)=\sigma !$,
• $\Gamma \left(1\right)=1$.

Definition 2 

([26]). The beta function for $\Re \left(\sigma \right)>0$ and $\Re \left(\tau \right)>0$ is defined by

$$\begin{array}{c}\hfill B(\sigma ,\tau )={\int }_0^1{\omega }^{\sigma -1}{(1-\omega )}^{\tau -1}d\omega .\end{array}$$

The beta function satisfies the following properties:

• $B(\sigma ,\tau )=B(\tau ,\sigma )$,
• $B(\sigma ,\tau )={\displaystyle \frac{\Gamma \left(\sigma \right)\Gamma \left(\tau \right)}{\Gamma (\sigma +\tau )}}$.

Definition 3 

([26]). The Pochhammer symbol ${\left(\sigma \right)}_n$, where $\sigma \in \mathbb{C}$ and $n\in {\mathbb{N}}_0$, is defined by the relation

$$\begin{array}{c}\hfill {\left(\sigma \right)}_n={\displaystyle \frac{\Gamma (\sigma +n)}{\Gamma \left(\sigma \right)}}=\left\{\begin{array}{cc}\sigma (\sigma +1)\dots (\sigma +n-1),\hfill & n\in \mathbb{N},\hfill \\ 1,\hfill & n=0.\hfill \end{array}\right.\end{array}$$

The Pochhammer symbol satisfies the following properties:

• ${\left(\sigma \right)}_{m+n}={\left(\sigma \right)}_m{(\sigma +m)}_n$,
• ${\left(\sigma \right)}_{n+1}=\sigma {(\sigma +1)}_n$,
• ${\left(1\right)}_n=n!$.

Definition 4 

([27,28]). The classical Srivastava’s triple hypergeometric functions $H_A$, $H_B$ and $H_C$, respectively, are defined by

$$\begin{array}{c}\hfill H_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{\left({\mu }_2\right)}_{m+n}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_{n+k}}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}},\end{array}$$

$$\left(\mathbf{s}<1,\mathbf{t}<1,\mathbf{r}<(1-\mathbf{s})(1-\mathbf{t})\right),$$

$$\begin{array}{c}\hfill H_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{\left({\mu }_2\right)}_{m+n}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_n{\left({\mu }_6\right)}_k}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}},\end{array}$$

$$\left(\mathbf{r}+\mathbf{s}+\mathbf{t}+2\sqrt{\mathbf{r}\mathbf{s}\mathbf{t}}<1\right),$$

and

$$\begin{array}{c}\hfill H_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{\left({\mu }_2\right)}_{m+n}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_{m+n+k}}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}},\end{array}$$

$$\left(\mathbf{r}<1,\mathbf{s}<1,\mathbf{t}<1,\mathbf{r}+\mathbf{s}+\mathbf{t}-2\sqrt{(1-\mathbf{r})(1-\mathbf{s})(1-\mathbf{t})}<2\right),$$

where, for simplicity of notation, the coordinates $(\mathbf{r},\mathbf{s},\mathbf{t})$ are used in place of $\left(\right|\kappa |,|\omega |,|\tau \left|\right)$.

3. Extended Srivastava Hypergeometric Functions

In this section, we introduce new extensions of the classical Srivastava triple hypergeometric functions via the general kernel-based beta function. This construction provides a unified and flexible framework that includes several known extensions as special cases corresponding to appropriate choices of the kernel function. The proposed approach preserves the analytic structure of the classical Srivastava functions while enriching them through additional deformation parameters embedded in the kernel.

We first recall the general kernel-based beta function.

Definition 5 

([25]). The general kernel-based beta function is defined by

$$\begin{array}{c}\hfill ᵏ\widehat{B}(\sigma ,\tau )={\int }_0^1{\omega }^{\sigma -1}{(1-\omega )}^{\tau -1}K(\omega ,\mathbf{X})d\omega ,\end{array}$$

where $\mathbf{X}=\mathbf{X}(p,q,\alpha ,\beta )$ denotes a multi-parameter variable and $K(\omega ,\mathbf{X})$ denotes a general kernel function. The parameters satisfy

$$\Re \left(p\right)>0,\Re \left(q\right)>0,\Re \left(\alpha \right)>0,\Re \left(\beta \right)>0,$$

and

$$\Re \left(\sigma \right)>0,\Re \left(\tau \right)>0.$$

The general kernel function $K(\omega ,\mathbf{X})$ admits several important structures, allowing the general kernel-based beta function definition to unify a broad family of extended or generalized beta functions. In this context, the $K(\omega ,\mathbf{X})$ can take the following forms:

(i) Product-type kernels

• $K(\omega ,\mathbf{X})=K\left(-{\displaystyle \frac{p}{{\omega }^{\alpha }}}\right)K\left(-{\displaystyle \frac{q}{{(1-\omega )}^{\beta }}}\right)$,
• $K(\omega ,\mathbf{X})=K\left(-{\displaystyle \frac{p}{{\omega }^{\alpha }}}\right)K\left(-{\displaystyle \frac{q}{{(1-\omega )}^{\alpha }}}\right)$,
• $K(\omega ,\mathbf{X})=K\left(-{\displaystyle \frac{p}{\omega }}\right)K\left(-{\displaystyle \frac{q}{1-\omega }}\right)$.

(ii) Additive-type kernels

• $K(\omega ,\mathbf{X})=K\left(-{\displaystyle \frac{p}{{\omega }^{\alpha }}}-{\displaystyle \frac{q}{{(1-\omega )}^{\beta }}}\right)$,
• $K(\omega ,\mathbf{X})=K\left(-{\displaystyle \frac{p}{{\omega }^{\alpha }}}-{\displaystyle \frac{q}{{(1-\omega )}^{\alpha }}}\right)$,
• $K(\omega ,\mathbf{X})=K\left(-{\displaystyle \frac{p}{\omega }}-{\displaystyle \frac{q}{1-\omega }}\right)$.

(iii) Coupled-type kernels

• $K(\omega ,\mathbf{X})=K\left(-{\displaystyle \frac{p}{{\omega }^{\alpha }{(1-\omega )}^{\beta }}}\right)$,
• $K(\omega ,\mathbf{X})=K\left(-{\displaystyle \frac{p}{{\omega }^{\alpha }{(1-\omega )}^{\alpha }}}\right)$,
• $K(\omega ,\mathbf{X})=K\left(-{\displaystyle \frac{p}{\omega (1-\omega )}}\right)$.

The general kernel function K may be selected from a broad class of special functions such as the exponential function, Kummer confluent hypergeometric function, Mittag–Leffler function, Wright function, Fox–Wright function, and the M-series. Consequently, the proposed framework provides a unified representation for many extended or generalized beta-type functions available in the literature.

Definition 6. 

Assume that the coordinates are given by $(\mathbf{r},\mathbf{s},\mathbf{t})$ instead of $\left(\right|\kappa |,|\omega |,|\tau \left|\right)$ and let the parameters satisfy $\Re \left(p\right)>0,\Re \left(q\right)>0,\Re \left(\alpha \right)>0,\Re \left(\beta \right)>0$. Then, using the general kernel-based beta function, the extended forms of the Srivastava hypergeometric functions $H_A$, $H_B$, and $H_C$ are defined as follows:

$$\begin{array}{cc}\hfill ᵏ{\widehat{H}}_A& ({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\hfill \\ \hfill & =ᵏH_{A,p,q}^{(\alpha ,\beta )}({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{\left({\mu }_2\right)}_{m+n}}{{\left({\mu }_4\right)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+n+k,{\mu }_5-{\mu }_3)}{B({\mu }_3,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}},\hfill \end{array}$$

$$\left(\mathbf{s}<1,\mathbf{t}<1,\mathbf{r}<(1-\mathbf{s})(1-\mathbf{t})\right),$$

$$\begin{array}{cc}\hfill ᵏ{\widehat{H}}_B& ({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\hfill \\ \hfill & =ᵏH_{B,p,q}^{(\alpha ,\beta )}({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2)}_{2m+n+k}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_n{\left({\mu }_6\right)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_2+m+n)}{B({\mu }_1,{\mu }_2)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}},\hfill \end{array}$$

$$\left(\mathbf{r}+\mathbf{s}+\mathbf{t}+2\sqrt{\mathbf{r}\mathbf{s}\mathbf{t}}<1\right),$$

and

$$\begin{array}{cc}\hfill ᵏ{\widehat{H}}_C& ({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\hfill \\ \hfill & =ᵏH_{C,p,q}^{(\alpha ,\beta )}({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_4+n-{\mu }_1)}{B({\mu }_1,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}},\hfill \end{array}$$

$$\left(\mathbf{r}<1,\mathbf{s}<1,\mathbf{t}<1,\mathbf{r}+\mathbf{s}+\mathbf{t}-2\sqrt{(1-\mathbf{r})(1-\mathbf{s})(1-\mathbf{t})}<2\right).$$

They are called the Srivastava hypergeometric function $ᵏ{\widehat{H}}_A$, the Srivastava hypergeometric function $ᵏ{\widehat{H}}_B$, and the Srivastava hypergeometric function $ᵏ{\widehat{H}}_C$, respectively. These functions, owing to the incorporation of the general kernel-based beta function, provide a unified framework that encompasses both the classical Srivastava functions and various extensions of Srivastava-type hypergeometric functions reported in the literature.

4. Higher-Order Partial DIfferentiation Formulas

In this section, we derive a family of r-th order differentiation formulas for the newly introduced extended Srivastava hypergeometric functions $ᵏ{\widehat{H}}_A$, $ᵏ{\widehat{H}}_B$, and $ᵏ{\widehat{H}}_C$. The main objective is to investigate how these functions behave under repeated partial differentiation with respect to the parameters $\kappa$, $\omega$, and $\tau$. By working directly from the defining triple series representations, we apply term-by-term differentiation and use classical identities for factorials, the gamma function, and Pochhammer symbols. This procedure reveals that each differentiation preserves the structural form of the original function, while only shifting the involved parameters. As a result, all obtained formulas exhibit a unified pattern: the r-th derivative of each function can be expressed as a scaled version of the same function with appropriately shifted parameters. This demonstrates that the introduced function families are closed under differentiation and possess a stable algebraic structure, which is essential for their applications in fractional calculus and related differential equation models.

Theorem 1. 

For $r\in {\mathbb{N}}_0$, the following partial differentiation formula with respect to κ holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}\left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_2\right)}_r}{{\left({\mu }_4\right)}_r}}ᵏ{\widehat{H}}_A({\mu }_1+r,{\mu }_2+r,{\mu }_3;{\mu }_4+r,{\mu }_5;\kappa ,\omega ,\tau ).\hfill \end{array}$$

Proof. 

Starting from the defining series representation

$$\begin{array}{cc}\hfill ᵏ{\widehat{H}}_A& ({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{\left({\mu }_2\right)}_{m+n}}{{\left({\mu }_4\right)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+n+k,{\mu }_5-{\mu }_3)}{B({\mu }_3,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}},\hfill \end{array}$$

we differentiate term-by-term with respect to $\kappa$. Thus,

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}& \left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{\left({\mu }_2\right)}_{m+n}}{{\left({\mu }_4\right)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+n+k,{\mu }_5-{\mu }_3)}{B({\mu }_3,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}{\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}\left({\displaystyle \frac{{\kappa }^m}{m!}}\right).\hfill \end{array}$$

Using

$${\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}\left({\displaystyle \frac{{\kappa }^m}{m!}}\right)={\displaystyle \frac{{\kappa }^{m-r}}{(m-r)!}},m\ge r,$$

we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}& \left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m=r}^{\infty }}{\displaystyle \sum _{n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{\left({\mu }_2\right)}_{m+n}}{{\left({\mu }_4\right)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+n+k,{\mu }_5-{\mu }_3)}{B({\mu }_3,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^{m-r}}{(m-r)!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Now, by shifting the index $m\mapsto m+r$, we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}& \left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+r+k}{\left({\mu }_2\right)}_{m+r+n}}{{\left({\mu }_4\right)}_{m+r}}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+n+k,{\mu }_5-{\mu }_3)}{B({\mu }_3,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Next, applying the Pochhammer decomposition formula

$${\left(a\right)}_{m+r}={\left(a\right)}_r{(a+r)}_m,$$

we obtain

$${\left({\mu }_1\right)}_{m+r+k}={\left({\mu }_1\right)}_r{({\mu }_1+r)}_{m+k},$$

$${\left({\mu }_2\right)}_{m+r+n}={\left({\mu }_2\right)}_r{({\mu }_2+r)}_{m+n},$$

and

$${\left({\mu }_4\right)}_{m+r}={\left({\mu }_4\right)}_r{({\mu }_4+r)}_m.$$

Substituting these relations into the above expression yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}& \left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_2\right)}_r}{{\left({\mu }_4\right)}_r}}{\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+r)}_{m+k}{({\mu }_2+r)}_{m+n}}{{({\mu }_4+r)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+n+k,{\mu }_5-{\mu }_3)}{B({\mu }_3,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Recognizing the resulting series as

$$ᵏ{\widehat{H}}_A({\mu }_1+r,{\mu }_2+r,{\mu }_3;{\mu }_4+r,{\mu }_5;\kappa ,\omega ,\tau ),$$

we finally obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}\left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_2\right)}_r}{{\left({\mu }_4\right)}_r}}ᵏ{\widehat{H}}_A({\mu }_1+r,{\mu }_2+r,{\mu }_3;{\mu }_4+r,{\mu }_5;\kappa ,\omega ,\tau ).\hfill \end{array}$$

This completes the proof. □

Theorem 2. 

For $r\in {\mathbb{N}}_0$, the following partial differentiation formula with respect to ω holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}\left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_2\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_5\right)}_r}}ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2+r,{\mu }_3+r;{\mu }_4,{\mu }_5+r;\kappa ,\omega ,\tau ).\hfill \end{array}$$

Proof. 

Starting from the defining series representation

$$\begin{array}{cc}\hfill ᵏ{\widehat{H}}_A& ({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{\left({\mu }_2\right)}_{m+n}}{{\left({\mu }_4\right)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+n+k,{\mu }_5-{\mu }_3)}{B({\mu }_3,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Differentiating term-by-term with respect to $\omega$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}& \left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{\left({\mu }_2\right)}_{m+n}}{{\left({\mu }_4\right)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+n+k,{\mu }_5-{\mu }_3)}{B({\mu }_3,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\tau }^k}{k!}}{\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}\left({\displaystyle \frac{{\omega }^n}{n!}}\right).\hfill \end{array}$$

Using

$${\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}\left({\displaystyle \frac{{\omega }^n}{n!}}\right)={\displaystyle \frac{{\omega }^{n-r}}{(n-r)!}},n\ge r,$$

we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}& \left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{n=r}^{\infty }}{\displaystyle \sum _{m,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{\left({\mu }_2\right)}_{m+n}}{{\left({\mu }_4\right)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+n+k,{\mu }_5-{\mu }_3)}{B({\mu }_3,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^{n-r}}{(n-r)!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Now, by shifting the index $n\mapsto n+r$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}& \left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{\left({\mu }_2\right)}_{m+n+r}}{{\left({\mu }_4\right)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+n+k+r,{\mu }_5-{\mu }_3)}{B({\mu }_3,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Applying the Pochhammer decomposition formula

$${\left(a\right)}_{m+r}={\left(a\right)}_r{(a+r)}_m,$$

we have

$${\left({\mu }_2\right)}_{m+n+r}={\left({\mu }_2\right)}_r{({\mu }_2+r)}_{m+n}.$$

Furthermore, using the beta-function identity

$$B({\mu }_3+r,{\mu }_5-{\mu }_3)={\displaystyle \frac{{\left({\mu }_3\right)}_r}{{\left({\mu }_5\right)}_r}}B({\mu }_3,{\mu }_5-{\mu }_3),$$

we obtain

$${\displaystyle \frac{1}{B({\mu }_3,{\mu }_5-{\mu }_3)}}={\displaystyle \frac{{\left({\mu }_3\right)}_r}{{\left({\mu }_5\right)}_r}}{\displaystyle \frac{1}{B({\mu }_3+r,{\mu }_5-{\mu }_3)}}.$$

Hence,

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}& \left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_2\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_5\right)}_r}}{\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{({\mu }_2+r)}_{m+n}}{{\left({\mu }_4\right)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+r+n+k,{\mu }_5-{\mu }_3)}{B({\mu }_3+r,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Recognizing the resulting series as

$$ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2+r,{\mu }_3+r;{\mu }_4,{\mu }_5+r;\kappa ,\omega ,\tau ),$$

we finally arrive at

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}\left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_2\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_5\right)}_r}}ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2+r,{\mu }_3+r;{\mu }_4,{\mu }_5+r;\kappa ,\omega ,\tau ).\hfill \end{array}$$

This completes the proof. □

Theorem 3. 

For $r\in {\mathbb{N}}_0$, the following partial differentiation formula with respect to τ holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}\left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_5\right)}_r}}ᵏ{\widehat{H}}_A({\mu }_1+r,{\mu }_2,{\mu }_3+r;{\mu }_4,{\mu }_5+r;\kappa ,\omega ,\tau ).\hfill \end{array}$$

Proof. 

Starting from the defining series representation

$$\begin{array}{cc}\hfill ᵏ{\widehat{H}}_A& ({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{\left({\mu }_2\right)}_{m+n}}{{\left({\mu }_4\right)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+n+k,{\mu }_5-{\mu }_3)}{B({\mu }_3,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Differentiating term-by-term with respect to $\tau$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}& \left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{\left({\mu }_2\right)}_{m+n}}{{\left({\mu }_4\right)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+n+k,{\mu }_5-{\mu }_3)}{B({\mu }_3,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}\left({\displaystyle \frac{{\tau }^k}{k!}}\right).\hfill \end{array}$$

Using

$${\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}\left({\displaystyle \frac{{\tau }^k}{k!}}\right)={\displaystyle \frac{{\tau }^{k-r}}{(k-r)!}},k\ge r,$$

we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}& \left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{k=r}^{\infty }}{\displaystyle \sum _{m,n=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k}{\left({\mu }_2\right)}_{m+n}}{{\left({\mu }_4\right)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+n+k,{\mu }_5-{\mu }_3)}{B({\mu }_3,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^{k-r}}{(k-r)!}}.\hfill \end{array}$$

Now, by shifting the index $k\mapsto k+r$, we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}& \left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_1\right)}_{m+k+r}{\left({\mu }_2\right)}_{m+n}}{{\left({\mu }_4\right)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+n+k+r,{\mu }_5-{\mu }_3)}{B({\mu }_3,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Applying the Pochhammer decomposition formula

$${\left(a\right)}_{m+r}={\left(a\right)}_r{(a+r)}_m,$$

we obtain

$${\left({\mu }_1\right)}_{m+k+r}={\left({\mu }_1\right)}_r{({\mu }_1+r)}_{m+k}.$$

Furthermore, using the beta-function identity

$$B({\mu }_3+r,{\mu }_5-{\mu }_3)={\displaystyle \frac{{\left({\mu }_3\right)}_r}{{\left({\mu }_5\right)}_r}}B({\mu }_3,{\mu }_5-{\mu }_3),$$

we have

$${\displaystyle \frac{1}{B({\mu }_3,{\mu }_5-{\mu }_3)}}={\displaystyle \frac{{\left({\mu }_3\right)}_r}{{\left({\mu }_5\right)}_r}}{\displaystyle \frac{1}{B({\mu }_3+r,{\mu }_5-{\mu }_3)}}.$$

Substituting these relations into the previous expression yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}& \left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_5\right)}_r}}{\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+r)}_{m+k}{\left({\mu }_2\right)}_{m+n}}{{\left({\mu }_4\right)}_m}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_3+r+n+k,{\mu }_5-{\mu }_3)}{B({\mu }_3+r,{\mu }_5-{\mu }_3)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Recognizing the resulting series as

$$ᵏ{\widehat{H}}_A({\mu }_1+r,{\mu }_2,{\mu }_3+r;{\mu }_4,{\mu }_5+r;\kappa ,\omega ,\tau ),$$

we finally obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}\left\{ᵏ{\widehat{H}}_A({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_5\right)}_r}}ᵏ{\widehat{H}}_A({\mu }_1+r,{\mu }_2,{\mu }_3+r;{\mu }_4,{\mu }_5+r;\kappa ,\omega ,\tau ).\hfill \end{array}$$

This completes the proof. □

Theorem 4. 

For $r\in {\mathbb{N}}_0$, the following partial differentiation formula with respect to κ holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}\left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_2\right)}_r}{{\left({\mu }_4\right)}_r}}ᵏ{\widehat{H}}_B({\mu }_1+r,{\mu }_2+r,{\mu }_3;{\mu }_4+r,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau ).\hfill \end{array}$$

Proof. 

Starting from the defining series representation

$$\begin{array}{cc}\hfill ᵏ{\widehat{H}}_B& ({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2)}_{2m+n+k}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_n{\left({\mu }_6\right)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_2+m+n)}{B({\mu }_1,{\mu }_2)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}},\hfill \end{array}$$

we differentiate term-by-term with respect to $\kappa$. Thus,

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}& \left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2)}_{2m+n+k}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_n{\left({\mu }_6\right)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_2+m+n)}{B({\mu }_1,{\mu }_2)}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}{\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}\left({\displaystyle \frac{{\kappa }^m}{m!}}\right).\hfill \end{array}$$

Using

$${\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}\left({\displaystyle \frac{{\kappa }^m}{m!}}\right)={\displaystyle \frac{{\kappa }^{m-r}}{(m-r)!}},m\ge r,$$

we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}& \left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m=r}^{\infty }}{\displaystyle \sum _{n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2)}_{2m+n+k}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_n{\left({\mu }_6\right)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_2+m+n)}{B({\mu }_1,{\mu }_2)}}{\displaystyle \frac{{\kappa }^{m-r}}{(m-r)!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Now, by shifting the index $m\mapsto m+r$, we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}& \left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2)}_{2m+2r+n+k}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_{m+r}{\left({\mu }_5\right)}_n{\left({\mu }_6\right)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+r+m+k,{\mu }_2+r+m+n)}{B({\mu }_1,{\mu }_2)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Next, using the Pochhammer decomposition formula

$${\left(a\right)}_{m+r}={\left(a\right)}_r{(a+r)}_m,$$

we obtain

$${\left({\mu }_4\right)}_{m+r}={\left({\mu }_4\right)}_r{({\mu }_4+r)}_m.$$

Furthermore, employing the beta-function identity

$$B(a+r,b+r)={\displaystyle \frac{{\left(a\right)}_r{\left(b\right)}_r}{{(a+b)}_{2r}}}B(a,b),$$

we deduce that

$${\displaystyle \frac{1}{B({\mu }_1,{\mu }_2)}}={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_2\right)}_r}{{({\mu }_1+{\mu }_2)}_{2r}}}{\displaystyle \frac{1}{B({\mu }_1+r,{\mu }_2+r)}}.$$

Substituting these relations into the previous expression yields

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}\left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_2\right)}_r}{{\left({\mu }_4\right)}_r}}{\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2+2r)}_{2m+n+k}{\left({\mu }_3\right)}_{n+k}}{{({\mu }_4+r)}_m{\left({\mu }_5\right)}_n{\left({\mu }_6\right)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+r+m+k,{\mu }_2+r+m+n)}{B({\mu }_1+r,{\mu }_2+r)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Finally, recognizing the resulting series as

$$ᵏ{\widehat{H}}_B({\mu }_1+r,{\mu }_2+r,{\mu }_3;{\mu }_4+r,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau ),$$

we arrive at

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}\left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_2\right)}_r}{{\left({\mu }_4\right)}_r}}ᵏ{\widehat{H}}_B({\mu }_1+r,{\mu }_2+r,{\mu }_3;{\mu }_4+r,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau ).\hfill \end{array}$$

This completes the proof. □

Theorem 5. 

For $r\in {\mathbb{N}}_0$, the following partial differentiation formula with respect to ω holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}\left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_2\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_5\right)}_r}}ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2+r,{\mu }_3+r;{\mu }_4,{\mu }_5+r,{\mu }_6;\kappa ,\omega ,\tau ).\hfill \end{array}$$

Proof. 

Starting from the defining series representation

$$\begin{array}{cc}\hfill ᵏ{\widehat{H}}_B& ({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2)}_{2m+n+k}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_n{\left({\mu }_6\right)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_2+m+n)}{B({\mu }_1,{\mu }_2)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}},\hfill \end{array}$$

we differentiate term-by-term with respect to $\omega$. Thus,

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}& \left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2)}_{2m+n+k}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_n{\left({\mu }_6\right)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_2+m+n)}{B({\mu }_1,{\mu }_2)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\tau }^k}{k!}}{\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}\left({\displaystyle \frac{{\omega }^n}{n!}}\right).\hfill \end{array}$$

Using

$${\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}\left({\displaystyle \frac{{\omega }^n}{n!}}\right)={\displaystyle \frac{{\omega }^{n-r}}{(n-r)!}},n\ge r,$$

we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}& \left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{n=r}^{\infty }}{\displaystyle \sum _{m,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2)}_{2m+n+k}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_n{\left({\mu }_6\right)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_2+m+n)}{B({\mu }_1,{\mu }_2)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^{n-r}}{(n-r)!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Now, by shifting the index $n\mapsto n+r$, we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}& \left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2)}_{2m+n+r+k}{\left({\mu }_3\right)}_{n+r+k}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_{n+r}{\left({\mu }_6\right)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_2+m+n+r)}{B({\mu }_1,{\mu }_2)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Next, applying the Pochhammer decomposition formula

$${\left(a\right)}_{m+r}={\left(a\right)}_r{(a+r)}_m,$$

we obtain

$${\left({\mu }_3\right)}_{n+r+k}={\left({\mu }_3\right)}_r{({\mu }_3+r)}_{n+k},$$

and

$${\left({\mu }_5\right)}_{n+r}={\left({\mu }_5\right)}_r{({\mu }_5+r)}_n.$$

Furthermore, employing the beta-function identity

$$B(a,b+r)={\displaystyle \frac{{\left(b\right)}_r}{{(a+b)}_r}}B(a,b),$$

we deduce that

$${\displaystyle \frac{1}{B({\mu }_1,{\mu }_2)}}={\displaystyle \frac{{\left({\mu }_2\right)}_r}{{({\mu }_1+{\mu }_2)}_r}}{\displaystyle \frac{1}{B({\mu }_1,{\mu }_2+r)}},$$

and hence the factor ${({\mu }_1+{\mu }_2)}_r$ cancels. Therefore,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}\left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_2\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_5\right)}_r}}{\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2+r)}_{2m+n+k}{({\mu }_3+r)}_{n+k}}{{\left({\mu }_4\right)}_m{({\mu }_5+r)}_n{\left({\mu }_6\right)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_2+r+m+n)}{B({\mu }_1,{\mu }_2+r)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Recognizing the resulting series as

$$ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2+r,{\mu }_3+r;{\mu }_4,{\mu }_5+r,{\mu }_6;\kappa ,\omega ,\tau ),$$

we arrive at

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}\left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_2\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_5\right)}_r}}ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2+r,{\mu }_3+r;{\mu }_4,{\mu }_5+r,{\mu }_6;\kappa ,\omega ,\tau ).\hfill \end{array}$$

This completes the proof. □

Theorem 6. 

For $r\in {\mathbb{N}}_0$, the following partial differentiation formula with respect to τ holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}\left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_6\right)}_r}}ᵏ{\widehat{H}}_B({\mu }_1+r,{\mu }_2,{\mu }_3+r;{\mu }_4,{\mu }_5,{\mu }_6+r;\kappa ,\omega ,\tau ).\hfill \end{array}$$

Proof. 

Starting from the defining series representation

$$\begin{array}{cc}\hfill ᵏ{\widehat{H}}_B& ({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2)}_{2m+n+k}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_n{\left({\mu }_6\right)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_2+m+n)}{B({\mu }_1,{\mu }_2)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}},\hfill \end{array}$$

we differentiate term-by-term with respect to $\tau$. Thus,

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}& \left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2)}_{2m+n+k}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_n{\left({\mu }_6\right)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_2+m+n)}{B({\mu }_1,{\mu }_2)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}\left({\displaystyle \frac{{\tau }^k}{k!}}\right).\hfill \end{array}$$

Using

$${\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}\left({\displaystyle \frac{{\tau }^k}{k!}}\right)={\displaystyle \frac{{\tau }^{k-r}}{(k-r)!}},k\ge r,$$

we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}& \left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{k=r}^{\infty }}{\displaystyle \sum _{m,n=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2)}_{2m+n+k}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_n{\left({\mu }_6\right)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_2+m+n)}{B({\mu }_1,{\mu }_2)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^{k-r}}{(k-r)!}}.\hfill \end{array}$$

Now, by shifting the index $k\mapsto k+r$, we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}& \left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2)}_{2m+n+k+r}{\left({\mu }_3\right)}_{n+k+r}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_n{\left({\mu }_6\right)}_{k+r}}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k+r,{\mu }_2+m+n)}{B({\mu }_1,{\mu }_2)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Next, applying the Pochhammer decomposition formula

$${\left(a\right)}_{m+r}={\left(a\right)}_r{(a+r)}_m,$$

we obtain

$${\left({\mu }_3\right)}_{n+k+r}={\left({\mu }_3\right)}_r{({\mu }_3+r)}_{n+k},$$

and

$${\left({\mu }_6\right)}_{k+r}={\left({\mu }_6\right)}_r{({\mu }_6+r)}_k.$$

Furthermore, using

$${\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k+r,{\mu }_2+m+n)}{B({\mu }_1,{\mu }_2)}}={\displaystyle \frac{{\left({\mu }_1\right)}_r}{{({\mu }_1+{\mu }_2)}_r}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+r+m+k,{\mu }_2+m+n)}{B({\mu }_1+r,{\mu }_2)}},$$

the factor ${({\mu }_1+{\mu }_2)}_r$ is cancelled by

$${({\mu }_1+{\mu }_2)}_{2m+n+k+r}={({\mu }_1+{\mu }_2)}_r{({\mu }_1+{\mu }_2+r)}_{2m+n+k}.$$

Hence,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}\left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_6\right)}_r}}{\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_1+{\mu }_2+r)}_{2m+n+k}{({\mu }_3+r)}_{n+k}}{{\left({\mu }_4\right)}_m{\left({\mu }_5\right)}_n{({\mu }_6+r)}_k}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+r+m+k,{\mu }_2+m+n)}{B({\mu }_1+r,{\mu }_2)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Recognizing the resulting series as

$$ᵏ{\widehat{H}}_B({\mu }_1+r,{\mu }_2,{\mu }_3+r;{\mu }_4,{\mu }_5,{\mu }_6+r;\kappa ,\omega ,\tau ),$$

we arrive at

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}\left\{ᵏ{\widehat{H}}_B({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4,{\mu }_5,{\mu }_6;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_6\right)}_r}}ᵏ{\widehat{H}}_B({\mu }_1+r,{\mu }_2,{\mu }_3+r;{\mu }_4,{\mu }_5,{\mu }_6+r;\kappa ,\omega ,\tau ).\hfill \end{array}$$

This completes the proof. □

Theorem 7. 

For $r\in {\mathbb{N}}_0$, the following partial differentiation formula with respect to κ holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}\left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_2\right)}_r}{{\left({\mu }_4\right)}_r}}ᵏ{\widehat{H}}_C({\mu }_1+r,{\mu }_2+r,{\mu }_3;{\mu }_4+r;\kappa ,\omega ,\tau ).\hfill \end{array}$$

Proof. 

Starting from the defining series representation

$$\begin{array}{cc}\hfill ᵏ{\widehat{H}}_C& ({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_4+n-{\mu }_1)}{B({\mu }_1,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Differentiating term-by-term with respect to $\kappa$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}& \left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_4+n-{\mu }_1)}{B({\mu }_1,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}{\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}\left({\displaystyle \frac{{\kappa }^m}{m!}}\right).\hfill \end{array}$$

Using

$${\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}\left({\displaystyle \frac{{\kappa }^m}{m!}}\right)={\displaystyle \frac{{\kappa }^{m-r}}{(m-r)!}},m\ge r,$$

we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}& \left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m=r}^{\infty }}{\displaystyle \sum _{n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_4+n-{\mu }_1)}{B({\mu }_1,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^{m-r}}{(m-r)!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Now, by shifting the index $m\mapsto m+r$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}& \left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n+r}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k+r,{\mu }_4+n-{\mu }_1)}{B({\mu }_1,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Applying the Pochhammer decomposition formula

$${\left(a\right)}_{m+r}={\left(a\right)}_r{(a+r)}_m,$$

we have

$${\left({\mu }_2\right)}_{m+n+r}={\left({\mu }_2\right)}_r{({\mu }_2+r)}_{m+n}.$$

Furthermore, using the beta-function identity

$$B(a+r,b)={\displaystyle \frac{{\left(a\right)}_r}{{(a+b)}_r}}B(a,b),$$

we obtain

$$B({\mu }_1+r,{\mu }_4+n-{\mu }_1)={\displaystyle \frac{{\left({\mu }_1\right)}_r}{{({\mu }_4+n)}_r}}B({\mu }_1,{\mu }_4+n-{\mu }_1),$$

which yields

$${\displaystyle \frac{1}{B({\mu }_1,{\mu }_4+n-{\mu }_1)}}={\displaystyle \frac{{\left({\mu }_1\right)}_r}{{({\mu }_4+n)}_r}}{\displaystyle \frac{1}{B({\mu }_1+r,{\mu }_4+n-{\mu }_1)}}.$$

Since

$${({\mu }_4+n)}_r={\displaystyle \frac{{\left({\mu }_4\right)}_{n+r}}{{\left({\mu }_4\right)}_n}},$$

we have

$${\displaystyle \frac{1}{{\left({\mu }_4\right)}_n}}{\displaystyle \frac{1}{{({\mu }_4+n)}_r}}={\displaystyle \frac{1}{{\left({\mu }_4\right)}_{n+r}}}.$$

Hence,

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}& \left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_2\right)}_r}{{\left({\mu }_4\right)}_r}}{\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_2+r)}_{m+n}{\left({\mu }_3\right)}_{n+k}}{{({\mu }_4+r)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+r+m+k,{\mu }_4+n-{\mu }_1)}{B({\mu }_1+r,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Recognizing the resulting series as

$$ᵏ{\widehat{H}}_C({\mu }_1+r,{\mu }_2+r,{\mu }_3;{\mu }_4+r;\kappa ,\omega ,\tau ),$$

we finally obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\kappa }^r}}\left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_2\right)}_r}{{\left({\mu }_4\right)}_r}}ᵏ{\widehat{H}}_C({\mu }_1+r,{\mu }_2+r,{\mu }_3;{\mu }_4+r;\kappa ,\omega ,\tau ).\hfill \end{array}$$

This completes the proof. □

Theorem 8. 

For $r\in {\mathbb{N}}_0$, the following partial differentiation formula with respect to ω holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}\left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_2\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_4\right)}_r}}ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2+r,{\mu }_3+r;{\mu }_4+r;\kappa ,\omega ,\tau ).\hfill \end{array}$$

Proof. 

Starting from the defining series representation

$$\begin{array}{cc}\hfill ᵏ{\widehat{H}}_C& ({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_4+n-{\mu }_1)}{B({\mu }_1,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Differentiating term-by-term with respect to $\omega$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}& \left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_4+n-{\mu }_1)}{B({\mu }_1,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\tau }^k}{k!}}{\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}\left({\displaystyle \frac{{\omega }^n}{n!}}\right).\hfill \end{array}$$

Using

$${\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}\left({\displaystyle \frac{{\omega }^n}{n!}}\right)={\displaystyle \frac{{\omega }^{n-r}}{(n-r)!}},n\ge r,$$

we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}& \left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{n=r}^{\infty }}{\displaystyle \sum _{m,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_4+n-{\mu }_1)}{B({\mu }_1,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^{n-r}}{(n-r)!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Now, by shifting the index $n\mapsto n+r$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}& \left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n+r}{\left({\mu }_3\right)}_{n+k+r}}{{\left({\mu }_4\right)}_{n+r}}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_4+n+r-{\mu }_1)}{B({\mu }_1,{\mu }_4+n+r-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Applying the Pochhammer decomposition formula

$${\left(a\right)}_{m+r}={\left(a\right)}_r{(a+r)}_m,$$

we have

$${\left({\mu }_2\right)}_{m+n+r}={\left({\mu }_2\right)}_r{({\mu }_2+r)}_{m+n},$$

and

$${\left({\mu }_3\right)}_{n+k+r}={\left({\mu }_3\right)}_r{({\mu }_3+r)}_{n+k}.$$

Also,

$${\left({\mu }_4\right)}_{n+r}={\left({\mu }_4\right)}_r{({\mu }_4+r)}_n.$$

Hence,

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}& \left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_2\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_4\right)}_r}}{\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{({\mu }_2+r)}_{m+n}{({\mu }_3+r)}_{n+k}}{{({\mu }_4+r)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_4+r+n-{\mu }_1)}{B({\mu }_1,{\mu }_4+r+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Recognizing the resulting series as

$$ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2+r,{\mu }_3+r;{\mu }_4+r;\kappa ,\omega ,\tau ),$$

we finally obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\omega }^r}}\left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_2\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_4\right)}_r}}ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2+r,{\mu }_3+r;{\mu }_4+r;\kappa ,\omega ,\tau ).\hfill \end{array}$$

This completes the proof. □

Theorem 9. 

For $r\in {\mathbb{N}}_0$, the following partial differentiation formula with respect to τ holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}\left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_4\right)}_r}}ᵏ{\widehat{H}}_C({\mu }_1+r,{\mu }_2,{\mu }_3+r;{\mu }_4+r;\kappa ,\omega ,\tau ).\hfill \end{array}$$

Proof. 

Starting from the defining series representation

$$\begin{array}{cc}\hfill ᵏ{\widehat{H}}_C& ({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_4+n-{\mu }_1)}{B({\mu }_1,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Differentiating term-by-term with respect to $\tau$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}& \left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_4+n-{\mu }_1)}{B({\mu }_1,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}\left({\displaystyle \frac{{\tau }^k}{k!}}\right).\hfill \end{array}$$

Using

$${\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}\left({\displaystyle \frac{{\tau }^k}{k!}}\right)={\displaystyle \frac{{\tau }^{k-r}}{(k-r)!}},k\ge r,$$

we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}& \left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{k=r}^{\infty }}{\displaystyle \sum _{m,n=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n}{\left({\mu }_3\right)}_{n+k}}{{\left({\mu }_4\right)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k,{\mu }_4+n-{\mu }_1)}{B({\mu }_1,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^{k-r}}{(k-r)!}}.\hfill \end{array}$$

Now, by shifting the index $k\mapsto k+r$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}& \left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n}{\left({\mu }_3\right)}_{n+k+r}}{{\left({\mu }_4\right)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+m+k+r,{\mu }_4+n-{\mu }_1)}{B({\mu }_1,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Applying the Pochhammer decomposition formula

$${\left(a\right)}_{m+r}={\left(a\right)}_r{(a+r)}_m,$$

we have

$${\left({\mu }_3\right)}_{n+k+r}={\left({\mu }_3\right)}_r{({\mu }_3+r)}_{n+k}.$$

Furthermore, using the beta-function identity

$$B(a+r,b)={\displaystyle \frac{{\left(a\right)}_r}{{(a+b)}_r}}B(a,b),$$

we obtain

$$B({\mu }_1+r,{\mu }_4+n-{\mu }_1)={\displaystyle \frac{{\left({\mu }_1\right)}_r}{{({\mu }_4+n)}_r}}B({\mu }_1,{\mu }_4+n-{\mu }_1).$$

Hence,

$${\displaystyle \frac{1}{B({\mu }_1,{\mu }_4+n-{\mu }_1)}}={\displaystyle \frac{{\left({\mu }_1\right)}_r}{{({\mu }_4+n)}_r}}{\displaystyle \frac{1}{B({\mu }_1+r,{\mu }_4+n-{\mu }_1)}}.$$

Since

$${({\mu }_4+n)}_r={\displaystyle \frac{{\left({\mu }_4\right)}_{n+r}}{{\left({\mu }_4\right)}_n}},$$

we get

$${\displaystyle \frac{1}{{({\mu }_4+n)}_r}}={\displaystyle \frac{{\left({\mu }_4\right)}_n}{{\left({\mu }_4\right)}_{n+r}}}.$$

Substituting these relations into the previous expression yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}& \left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\left({\mu }_1\right)}_r{\left({\mu }_3\right)}_r{\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n}{({\mu }_3+r)}_{n+k}}{{\left({\mu }_4\right)}_{n+r}}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+r+m+k,{\mu }_4+n-{\mu }_1)}{B({\mu }_1+r,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Also, we have

$${\displaystyle \frac{1}{{\left({\mu }_4\right)}_{n+r}}}={\displaystyle \frac{1}{{\left({\mu }_4\right)}_r{({\mu }_4+r)}_n}}.$$

Then, we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}& \left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_4\right)}_r}}{\displaystyle \sum _{m,n,k=0}^{\infty }}{\displaystyle \frac{{\left({\mu }_2\right)}_{m+n}{({\mu }_3+r)}_{n+k}}{{({\mu }_4+r)}_n}}{\displaystyle \frac{ᵏ\widehat{B}({\mu }_1+r+m+k,{\mu }_4+n-{\mu }_1)}{B({\mu }_1+r,{\mu }_4+n-{\mu }_1)}}{\displaystyle \frac{{\kappa }^m}{m!}}{\displaystyle \frac{{\omega }^n}{n!}}{\displaystyle \frac{{\tau }^k}{k!}}.\hfill \end{array}$$

Recognizing the resulting series as

$$ᵏ{\widehat{H}}_C({\mu }_1+r,{\mu }_2,{\mu }_3+r;{\mu }_4+r;\kappa ,\omega ,\tau ),$$

we finally obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^r}{\partial {\tau }^r}}\left\{ᵏ{\widehat{H}}_C({\mu }_1,{\mu }_2,{\mu }_3;{\mu }_4;\kappa ,\omega ,\tau )\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left({\mu }_1\right)}_r{\left({\mu }_3\right)}_r}{{\left({\mu }_4\right)}_r}}ᵏ{\widehat{H}}_C({\mu }_1+r,{\mu }_2,{\mu }_3+r;{\mu }_4+r;\kappa ,\omega ,\tau ).\hfill \end{array}$$

This completes the proof. □

5. Results and Discussion

In this section, we summarize the main results obtained in the previous part and discuss their structural and analytical implications for the extended Srivastava-type hypergeometric functions $ᵏ{\widehat{H}}_A$, $ᵏ{\widehat{H}}_B$, and $ᵏ{\widehat{H}}_C$.

Results

We have derived explicit r-th order partial differentiation formulas with respect to the variables $\kappa$, $\omega$, and $\tau$ for each class of the introduced functions. In all cases, the following common structural properties are observed:

• The r-th derivative with respect to a given variable preserves the original functional form.
• Each differentiation introduces multiplicative coefficients expressed in terms of Pochhammer symbols.
• The parameters of the function are systematically shifted by the order of differentiation r.

More precisely, for each family $ᵏ{\widehat{H}}_A$, $ᵏ{\widehat{H}}_B$, and $ᵏ{\widehat{H}}_C$, we obtain identities of the general form

$${\displaystyle \frac{{\partial }^r}{\partial x^r}}ᵏ\widehat{H}(·)=\left(\mathrm{Pochhammer}\mathrm{factors}\right)\times ᵏ\widehat{H}\left(\mathrm{shifted}\mathrm{parameters}\right),$$

where $x\in \{\kappa ,\omega ,\tau \}$.

Discussion

The obtained results show that all three function classes possess a strong closure property under partial differentiation. In particular, repeated differentiation does not alter the functional structure but only modifies the parameter set in a controlled way. This indicates a stable algebraic framework similar to classical hypergeometric functions.

From an analytical point of view, this property is significant for several reasons:

• It simplifies the treatment of higher-order differential equations involving these functions.
• It enables direct construction of recursive relations between derivatives.
• It provides a natural framework for applications in fractional calculus, where such shift-operator behavior is frequently encountered.

Moreover, the obtained parameter-shift structures appearing in the differentiation formulas indicate that the function families $ᵏ{\widehat{H}}_A$, $ᵏ{\widehat{H}}_B$, and $ᵏ{\widehat{H}}_C$ possess a common analytical framework. This demonstrates that the proposed kernel-based construction provides a unified framework for generalized Srivastava-type hypergeometric functions.

Overall, the results demonstrate that these functions are not only well-defined analytically but also highly structured under differentiation, making them suitable candidates for further study in integral transforms, fractional differential equations, and applied mathematical modeling.

6. Conclusions

In this work, we have established a systematic investigation of the differential properties of the extended Srivastava-type hypergeometric functions $ᵏ{\widehat{H}}_A$, $ᵏ{\widehat{H}}_B$, and $ᵏ{\widehat{H}}_C$ with respect to the variables $\kappa$, $\omega$, and $\tau$. By employing their series representations together with classical identities of the Pochhammer symbol and the beta function, we derived explicit r-th order differentiation formulas.

The obtained results show that differentiation with respect to each variable preserves the structural form of the underlying functions, while inducing systematic parameter shifts and multiplicative Pochhammer factors. This invariance-type property highlights the algebraic stability of the considered extended hypergeometric family under repeated differentiation.

Moreover, the presented formulas provide a unified framework that connects the different members of the $ᵏ\widehat{H}$-family, revealing consistent patterns in parameter evolution across $ᵏ{\widehat{H}}_A$, $ᵏ{\widehat{H}}_B$, and $ᵏ{\widehat{H}}_C$. These results may be useful in further analytical studies, particularly in solving fractional differential equations, constructing integral transforms, and investigating applied models in mathematical physics and engineering.

Future research may focus on extending these results to fractional-order derivatives, integral representations, and generating function approaches, as well as exploring potential applications in special function theory and fractional calculus.