Legendre-Gould Hopper based Sheﬀer polynomials: properties and applications

: In this article, the Legendre-Gould Hopper polynomials are combined with Sheﬀer sequences to introduce certain mixed type special polynomials. Generating functions, diﬀerential equations and certain other properties of Legendre-Gould Hopper based Sheﬀer polynomials are derived. Further, operational and integral representations provid-ing connections between these polynomials and known special polynomials are established. Certain identities and results for some members of these new mixed polynomials are also obtained. Finally, the determinantal deﬁnitions of Legendre-Gould Hopper based Sheﬀer polynomials are also given.


Introduction and preliminaries
The 0 th -order Tricomi function C 0 (x) is also given by the following operational definition: (1.5) where D −1 x denotes the inverse of the derivative operator D x := ∂ ∂x and Also, it is known that [16] − ∂ ∂x x ∂ ∂x C 0 (αx) = α C 0 (αx). (1.7) The LeGHP R H (s) n (x,y,z) n! and S H (s) n (x, y, z) are shown to be quasi-monomial [12,28] under the action of the following multiplicative and derivative operators [30]: Consequently,M RH ,P RH andM SH ,P SH satisfy the following recurrence relations: n+1 (x, y, z) (n + 1)! , (1.12) n (x, y, z) n! = n R H respectively, for all n ∈ N.
In view of the monomiality principle equationŝ n (x, y, z) are [30]:  n (x, y, z) give a number of other known special polynomials as special cases. We mention these special cases in Table 1.
Legendre [19] Rn(x, y) = (n!) 2 n r=0 y k z r x n−3k−2r k!r!(n−3k−2r)! Next, we recall that the polynomial sequence {s n (x)} ∞ n=0 (s n (x) being a polynomial of degree n) is called Sheffer A-type zero [26, p.222 (Theorem 72)], (which we shall hereafter call Sheffer-type), if s n (x) possesses the exponential generating function of the form where A(t) and H(t) have (at least the formal) expansions: and respectively.
Properties of Appell and Sheffer sequences are naturally handled within the framework of modern classical umbral calculus by Roman [27]. In view of the following result [27, p.17], the Sheffer sequences can be alternatively defined as: Let f (t) be a delta series and g(t) be an invertible series of the following form: and Then there exists a unique sequence s n (x) of polynomials satisfying the orthogonality conditions g(t)f (t) k |s n (x) = n! δ n,k , f or all n, k ≥ 0. (1.29) Also, in view of Roman [27,p.18 (Theorem 2.3.4)], the polynomial sequence s n (x) is uniquely determined by two (formal) power series given by equations (1.27) and (1.28). The exponential generating function of s n (x) is then given by and The sequence s n (x) in equation (1.29) is the Sheffer sequence for the pair (g(t), f (t)). The Sheffer sequence for (1, f (t)) is called the associated Sheffer sequence for f (t) and the Sheffer sequence for (g(t), t) becomes the Appell sequence for g(t) [27, p.17].
The Sheffer class contains very important sequences such as the Hermite, Laguerre, Bernoulli, Poisson-Charlier polynomials etc. These polynomials are important from the view point of applications in physics, number theory and in many other branches of mathematics. We present the lists of some known Sheffer and associated Sheffer families in Tables  2 and 3 respectively. Table 2. Some known Sheffer polynomials n (x) [1,26] III. ; ln 1+t

Mittag-Leffler
polynomials Mn(x) [3] II. [9,24] In the present paper, the Legendre-Gould Hopper based Sheffer polynomials are introduced and framed within the context of monomiality principle. Some operational and integral formulas for these polynomials are derived. Further, some results are obtained for some members of the Legendre-Gould Hopper based Sheffer polynomial families. The paper is also concluded with the determinantal definitions of the Legendre-Gould Hopper based Sheffer polynomials(LeGHSP) R H (s) s n (x, y, z) and S H (s) s n (x, y, z).

Legendre-Gould Hopper based Sheffer polynomials
To generate the Legendre-Gould Hopper based Sheffer polynomials (LeGHSP) denoted by R H (s) s n (x, y, z) and S H (s) s n (x, y, z), we prove the following results: Theorem 2.1. The Legendre-Gould Hopper based Sheffer polynomials (LeGHSP) denoted by R H (s) s n (x, y, z) and S H (s) s n (x, y, z) are defined by the following generating functions or, equivalently and or, equivalently respectively.
Proof. Replacing x in the l.h.s. and r.h.s. of equation (1.30) by the multiplicative operator Using the expression ofM RH given in equation (1.8) and then decoupling the exponential operator in the l.h.s. of the resultant equation by using the Crofton-type identity [18, p.12] (2.7) Now, using the Weyl identity [18] (2.9) Now, expanding the first exponential in the l.h.s. of equation (2.9) and using definition (1.5), we find Finally, denoting the resultant LeGHSP in the r.h.s. by R H (s) s n (x, y, z), that is we get assertion (2.1). Also, in view of equations (1.31) and (1.32), generating function (2.1) can be expressed equivalently as equation (2.2). Making use of (1.10) and using a similar argument as in the above proof of (2.1), we can establish the assertions (2.3) and (2.4).
Next, to show that the LeGHSP R H (s) s n (x, y, z) and S H (s) s n (x, y, z) satisfy the monomiality property, we prove the following result: Theorem 2.2. The Legendre-Gould Hopper based Sheffer polynomials R H (s) s n (x, y, z) and S H (s) s n (x, y, z) are quasi-monomial with respect to the following multiplicative and derivative operators:M , (2.13) or, equivalentlŷ (2.14) andP RHs := f (D y yD y ), (2.16) P SHs := f (D y ), (2.17) or, equivalentlyP respectively.
Proof. Consider the following identity: (2.20) Since f −1 denotes the compositional inverse of the function f and f (t) has an expansion (1.27) in powers of t, therefore we have (2.21) Differentiating equation (2.5) partially with respect to t and in view of relation (2.11), we find Since g(t) is an invertible series and f (t) is a delta series of t therefore possess power series expansions of f −1 (t). Thus, in view of relation (2.20), the above equation becomes which on using generating function (2.1) becomes (2.25) Adjusting the summation in the l.h.s. of the above equation and then equating the coefficients of like powers of t, we find which, in view of equation (1.12) shows that the multiplicative operator for R H (s) s n (x, y, z) is given as:M Finally, using equation (1.8) in the r.h.s of above equation, we get assertion (2.12). Next, consider the following identity and by making use of (1.10) and using a similar argument as in the above proof of (2.12), we establish the assertion (2.13). Again, in view of identity (2.21), we have which on using generating function (2.1) becomes Adjusting the summation in the l.h.s. of the above equation and then equating the coefficients of like powers of t, we get respectively.
Theorem 2.3. The Legendre-Gould Hopper based Sheffer polynomials R H (s) s n (x, y, z) and S H (s) s n (x, y, z) are the solutions of the following differential equations: and Remark 2.1. Since, for g(t) = 1 (or A(t) = 1), the Sheffer polynomials s n (x) reduce to the associated Sheffer polynomials s n (x). Therefore, taking g(t) = 1 (or A(t) = 1) in the results obtained in Theorems 2.1-2.3 and denoting the resultant Legendre-Gould Hopper based associated Sheffer polynomials (LeGHASP) by R H (s) s n (x, y, z) and S H (s) s n (x, y, z), we deduce the following consequences of Theorems 2.1-2.3 : Corollary 2.2. The Legendre-Gould Hopper based associated Sheffer polynomials (LeGHASP) denoted by R H (s) s n (x, y, z) and S H (s) s n (x, y, z) are defined by the following generating functions  respectively.
We have mentioned special cases of the LeGHP S H (s) n (x, y, z) and R H (s) n (x,y,z) n! in Table  1. Now, for the same choice of the variables and indices the LeGHSP R H (s) s n (x, y, z) and S H (s) s n (x, y, z) reduce to the corresponding special cases. We mention these known and new special polynomials related to the Sheffer sequences in Table 4. Table 4. Special cases of LeGHSP R H (s) s n (x, y, z) and S H (s) s n (x, y, z) i  Table 4, the results for the special polynomials related to the Sheffer sequences can be obtained.
Next, we derive certain operational representations for the LeGHSP R H (s) s n (x, y, z) and S H (s) s n (x, y, z).

Operational and integral representations
To establish the operational representation for the LeGHSP R H (s) s n (x, y, z) and S H (s) s n (x, y, z), we prove the following results: respectively.
Proof. In view of equation (2.11), the proof of (3.1) is direct use of identity (2.6) and similarly the proof of (3.2) can be obtained.
Theorem 3.2. The following operational representation between the LeGHSP R H (s) s n (x, y, z) and the 2VLeSP R s n (x, y) holds true: Proof. From equation (2.2), we have Since, in view of Table 1(VI), we have Consequently, from Table 4(VI), we have R H (s) s n (x, y, 0) = R s n (x, y).   (3.10) Since, in view of Solving equation (3.10) subject to initial condition (3.12), we get assertion (3.9).
Theorem 3.5. The following operational representation between the LeGHSP S H (s) s n (x, y, z) and the GHSP H (s) s n (y, z) hold true: Proof. From equations (1.5) and (2.4), we have (3.14) Since, in view of Solving equation (3.14) subject to initial condition (3.16), we get assertion (3.13).
Next, we prove the integral for the LeGHSP R H (s) s n (x, y, z) and S H (s) s n (x, y, z) in the form of following theorem: Theorem 3.6. The following integral representations for the LeGHSP R H (s) s n (x, y, z) and S H (s) s n (x, y, z) hold true: Next, on using the integral representation of LeGHP R H (s) n (x,y,z) n! [30]: (3.20) in the l.h.s of equation (3.19) and interchanging the sides, we have (3.21) or, equivalently Again, using equation (3.19) in the r.h.s of the above equation, we find Finally, equating the coefficient of like powers of t in both sides of the above equation, we get assertion (3.17). Similarly, we can get assertion (3.18) Corollary 3.2. The following operational representation between the LeGHASP R H (s) s n (x, y, z) and the 2VLeASP R s n (x, y) holds true: , y), (3.27) or, equivalently R H (s) s n (x, y, z) = exp z ∂ s ∂D −s y R s n (x, y).
(3.28) Corollary 3.3. The following operational representation between the LeGHASP S H (s) s n (x, y, z) and the 2VLeTASP 2 L s n (x, y) holds true: S H (s) s n (x, y, z) = exp z ∂ s ∂y s 2 L s n (x, y).

(3.29)
Corollary 3.4. The following operational representation between the LeGHASP R H (s) s n (x, y, z) and the 2VGLTASP [m] L s n (x, y) hold true: (3.30) Corollary 3.5. The following operational representation between the LeGHASP S H (s) s n (x, y, z) and the GHASP H (s) s n (y, z) hold true: Corollary 3.6. The following integral representations for the LeGHASP R H (s) s n (x, y, z) and S H (s) s n (x, y, z) hold true: respectively.

Examples
The Sheffer polynomials have been studied because of their remarkable applications not only in different branches of mathematics but also in physics. The Sheffer and associated Sheffer class contains a number of important special polynomials. In this section, some results for the corresponding members of the Legendre-Gould Hopper based Sheffer polynomial families are obtained. We consider the following examples:  Table 5. Results for the LeGHGHP R H (s) H n,r,v (x, y, z) Generating functions e −t r exp z(vt) s C 0 xvt) C 0 − yvt =  n (x, y, z): Table 7. Results for the LeGHGLP n! R H (s) L (ν) n (x, y, z) n (x, y, z) = 0 n (x, y),    .17), we get the following results for the LeGHPP R H (s) P n (x, y, z): Table 9. Results for the LeGHPP R H (s) P n (x, y, z) I.
Generating functions t 1−t exp z ln 1+t   Table 11. Results for the LeGHMLP R H (s) M n (x, y, z) Generating functions exp z ln 1+t   Table 13. Results for the LeGHEP R H (s) ϕ n (x, y, z)

I.
Generating functions exp z e t − 1 s C 0 x e t − 1 C 0 − y e t − 1 = Similarly, for the other members of the Sheffer and associated families (see Tables 2 and  3), there exist new special polynomials belonging to the LeGHSP and LeGHASP families respectively. The generating functions and other properties of these special polynomials can be obtained from the results derived in the second and third sections.

Concluding Remarks
A determinantal definition for the classical Bernoulli polynomials introduced by Costabile et.al. [10] has given a new approach to Bernoulli polynomials which was further extended to provide the determinantal definition of the Appell polynomials [11]. Recently, the determinantal definition of Appell sequences is extended to Sheffer sequences by using the theory of Riordan arrays [29].
The determinantal approach considered in [10,11,29] provides motivation to consider the determinantal form of the new families of special polynomials. In this section, we give the determinantal definitions for the Legendre-Gould Hopper based Sheffer polynomials R H (s) s n (x, y, z) and S H (s) s n (x, y, z) as: