In the field of delta-sigma modulators, reducing system power consumption without sacrificing accuracy has become a challenge. The summing circuit, as the main part of the cascade-of-integrator-in-feedforward delta-sigma modulator, consumes a significant amount of power in active summing. On the other hand, passive summing circuits have low power consumption but can cause attenuation in the summing results. This article proposes a low-power multi-bit delta-sigma modulator based on passive and attenuationless summation scheme. The summation circuit achieves multiplication of the voltage signal carried on the summation capacitor through bidirectional sampling technique, compensating for the inherent attenuation caused by passive summation, thus eliminating the need for active OTA to achieve perfect summation. A 2-order delta-sigma modulator based on 4-bit 1-order passive noise-shaping (NS) SAR quantizer is designed to verify the scheme. By combining a passive NS SAR as the 4-bit quantizer and the third-stage integrator of the system, and incorporating an energy-efficient cascode floating inverter amplifier (FIA) structure as the core active OTA, these elements have significantly enhanced the energy-efficiency of the modulator. Simulation results show that at a supply voltage of 1.2 V and a bandwidth of 20 kHz, the SNDR reaches 102.62 dB, power consumption is only 148.32 μW, and Schreier figure-of-merit (FoM) of SNDR is 183.92 dB.

We evaluate various sums involving the powers of Fibonacci and Lucas numbers.

We evaluate various binomial sums involving the powers of Fibonacci and Lucas numbers.

In this article, authors introduce a new class of degenerate Hermite poly-Genocchi polynomials and give some identities of these polynomials related to the Stirling numbers of the second kind. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions. These results extend some known summations and identities of degenerate Hermite poly-Bernoulli numbers and polynomials studied by Khan [8].

We say that a system {zmF(z)}m=0∞ is a Beurling system if F is an outer function. Beurling’s approximation theorem asserts that if F is an outer function from H2(D) then the system {zmF(z)}m=0∞ is complete in the space H2(D). We prove that a Beurling system with F∈Hp(D),1≤p<∞ is an M−bases in Hp(D) with an explicit dual system. Any function f∈Hp(D),1≤p<∞ can be expanded as a series by the system {zmF(z)}m=0∞. For different methods of summation we characterize outer functions F for which the expansion converges to f. Related results for weighted Hardy spaces in the unit disc are studied. Particularly we prove Rosenblum’s hypothesis.

We evaluate some binomial cubic Fibonacci sums.

We study various properties of the polygonal numbers; such as their recurrence relations; fundamental identities; weighted binomial and ordinary sums; partial sums and generating functions of their powers; and a continued fraction representation for them. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers; unlike what obtains in most literature.

We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.

In this paper, we introduce a new class of Hermite-Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we derive symmetric identities of Hermite-Fubini numbers and polynomials by using generating functions.

In this paper, we introduce a new class of Laguerre-based Hermite-Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we derive symmetric identities of Laguerre-based Hermite-Fubini numbers and polynomials by using generating functions.

The aim of this work is to study generalizations of the notion of mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the Weak Law of Large Numbers in the same way that the ordinary mean relates to the Strong Law. We propose a further generalization, also based on an improper integral, called the doubly weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions, We also consider generalizations arising from Abel-Feynman type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function.

We introduce a new kind of extended Hermite-based Frobenius type Eulerian polynomials and then derive diverse explicit and implicit summation equations including some symmetric formulas by utilizing series manupulation method. Multifarious summation formulas and identities given earlier for some well known polynomials such as Eulerian polynomials and Frobenius type Eulerian polynomials are generalized.

In this paper, we introduce a general family of Lagrange-based Apostol-type Hermite polynomials thereby unifying the Lagrange-based Apostol Hermite-Bernoulli and the Lagrange-based Apostol Hermite-Genocchi polynomials. We also define Lagrange-based Apostol Hermite-Euler polynomials via the generating function. In terms of these generalizations, we find new and useful relations between the unified family and the Apostol Hermite-Euler polynomials. We also derive their explicit representations and list some basic properties of each of them. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions.

In this paper, we define a (p,v)-extension of Hurwitz-Lerch Zeta function by considering an extension of beta function defined by Parmar et al. [J. Classical Anal. 11 (2017) 81–106]. We obtain its basic properties which include integral representations, Mellin transformation, derivative formulas and certain generating relations. Also, we establish the special cases of the main results.

We present a differential-calculus-based method which allows one to derive more identities from {\it any} given Fibonacci-Lucas identity containing a finite number of terms and having at least one free index. The strength of our method is that no additional information is required about the given original identity. The method readily extends to a generalized Fibonacci sequence.

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the Integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, inverse trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.

Pieron's and Chocholle’s seminal psychophysical work predicts that human response time to information relative to visual contrast and/or sound frequency decreases when contrast intensity or sound frequency increases. The goal of this study is to bring to the fore the ability of individuals to use visual contrast intensity and sound frequency in combination for faster perceptual decisions of relative depth (“nearer”) in planar (2D) object configurations on the basis of physical variations in luminance contrast. Computer controlled images with two abstract patterns of varying contrast intensity, one on the left and one on the right, preceded or not by a pure tone of varying frequency, were shown to healthy young humans in controlled experimental sequences. Their task (two-alternative forced-choice) was to decide as quickly as possible which of two patterns, the left or the right one, in a given image appeared to “stand out as if it were nearer” in terms of apparent (subjective) visual depth. The results show that the combinations of varying relative visual contrast with sounds of varying frequency exploited here produced an additive effect on choice response times in terms of facilitation, where a stronger visual contrast combined with a higher sound frequency produced shorter forced-choice response times. This new effect is predicted by cross-modal audio-visual probability summation.

The main objective of this paper is to introduce a further extension of extended (p, q)-beta function by considering two Mittag-Leffler function in the kernel. We investigate various properties of this newly defined beta function such as integral representations, summation formulas and Mellin transform. We define extended beta distribution and its mean, variance and moment generating function with the help of extension of beta function. Also, we establish an extension of extended (p, q)-hypergeometric and (p, q)-confluent hypergeometric functions by using the extension of beta function. Various properties of newly defined extended hypergeometric and confluent hypergeometric functions such as integral representations, Mellin transformations, differentiation formulas, transformation and summation formulas are investigated.

A new generalization of extended beta function and its various properties, integral representations and distribution are given in this paper. In addition, we establish the generalization of extended hypergeometric and con uent hypergeometric functions using the newly extended beta function. The properties these extended and con uent hypergeometric functions such as integral representations, Mellin transformations, dif- ferentiation formulas, transformation and summation formulas are also investigated.