Certain Properties of Apostol-Type Hermite-Based- Frobenius-Genocchi Polynomials

This paper is well designed to set-up some new identities related to generalized Apostol-type Hermite-based-Frobenius-Genocchi polynomials and by applying the generating functions, we derive some implicit summation formulae and symmetric identities. Further a relationship between Array-type polynomials, Apostol-type Bernoulli polynomials and generalized Apostol-type Frobenius-Genocchi polynomials is also established.

Non-Linear Synthesis of Spectra

This chapter turns away from the linear world of the Fourier transform and introduces the concepts related to non-linear operations as a means of spectral modification through waveshaping. The idea of non-linear functions applied to simple sinusoidal signals is explored from various perspectives. Closed-form summation formulae are first shown as examples of non-linear techniques. This is followed by a thorough discussion of phase and frequency modulation methods, themselves also shown to be based on the application of non-linear waveshaping methods. This is complemented by the techniques of phase distortion and the more general vector phase shaping algorithm. A look into adaptive forms of frequency modulation is followed by a complete study of polynomial and other forms of waveshaping.

Lemniscate-like constants and infinite series

We introduce lemniscate-like constants by “twisting” the standard series expansions of the classical lemniscate constants via harmonic-type factors in the summand. Closed-form evaluations for these constants are established, and are then utilized to construct alternate proofs of summation formulae obtained recently via coefficient-extraction techniques applied to Kummer’s classical hypergeometric identity.

On the three families of extended Laguerre-based Apostol-type polynomials

In this paper, we introduce a new class of generalized extended Laguerre-based Apostol-type-Bernoulli, Apostol-type-Euler and Apostoltype-Genocchi polynomials. These Apostol type polynomials are used to connect Fubini-Hermite and Bell-Hermite polynomials and to find new representations. We derive some implicit summation formulae and symmetric identities for these families of special functions by applying the generating functions.

Задача Холмгрена для эллиптического уравнения с сингулярными коэффициентами

In the present work, we investigate the Holmgren problem for an multidimensional elliptic equation with several singular coefficients. We use a fundamental solution of the equation, containing Lauricella’s hypergeometric function in many variables. Then using an «abc» method, the uniqueness for the solution of the Holmgren problem is proved. Applying a method of Green’s function, we are able to find the solution of the problem in an explicit form. Moreover, decomposition and summation formulae, formulae of differentiation and some adjacent relations for Lauricella’s hypergeometric functions in many variables were used in order to find the explicit solution for the formulated problem. В данной работе мы исследуем задачу Холмгрена для многомерного эллиптического уравнения с несколькими сингулярными коэффициентами. Мы используем фундаментальное решение уравнения, содержащее гипергеометрическую функцию Лауричеллы от многих переменных. Затем методом «abc» доказывается единственность решения проблемы Холмгрена. Применяя метод функции Грина, мы можем найти решение задачи в явном виде. Более того, формулы разложения и суммирования, формулы дифференцирования и некоторые смежные соотношения для гипергеометрических функций Лауричеллы от многих переменных были использованы для нахождения явного решения поставленной задачи.

Summation formulae involving Stirling and Lah numbers

By making use of the generating function method, we derive several summation formulae involving Stirling numbers and Lah numbers as well as other classical combinatorial numbers named after Bernoulli, Euler, Bell, Genocchi, Cauchy, Derangement, Harmonic, Fibonacci and Lucas.

A novel Kind of the multi-index Beta, Gauss, and confluent hypergeometric functions

This research article elaborates on a novel expansion of the beta function by using the multi-index Mittag-Leffler function. Here, we derive some basic properties of this new beta function and then present a new type of beta dispersal as an application of our proposed beta function. We also introduce a novel expansion of Gauss and confluent hypergeometric functions for our newly initiated beta function. Some important properties of our proposed hypergeometric functions (like integral representations, differential formulae, transformations formulae, summation formulae, and a generating relation) are also pointed out systematically.

Generalized Hermite- based Apostol- Bernoulli, Euler, Genocchi polynomials and their relations

In this paper, we have generalized Apostol-Hermite-Bernoullli polynomials, Apostol-Hermite-Euler polynomials and Apostol-Hermite-Genocchi polynomials. We have shown that there is an intimate connection between these polynomials and derived some implicit summation formulae by applying the generating functions.

A Parametric Kind of Fubini Polynomials of a Complex Variable

In this paper, we propose a parametric kind of Fubini polynomials by defining the two specific generating functions. We also investigate some analytical properties (for example, summation formulae, differential formulae and relationships with other well-known polynomials and numbers) for our introduced polynomials in a systematic way. Furthermore, we consider some relationships for parametric kind of Fubini polynomials associated with Bernoulli, Euler, and Genocchi polynomials and Stirling numbers of the second kind.