Several Higher Differentiation Formulas of Special Functions

Several Differentiation Formulas of Special Functions

Several Integrability Formulas of Special Functions. Part II

In this article, we give several differentiation and integrability formulas of special and composite functions including the trigonometric function, the hyperbolic function and the polynomial function [3]. For simplicity, we adopt the following rules: r, x, a, b denote real numbers, n, m denote elements of N, A denotes a closed-interval subset of R, and Z denotes an open subset of R. One can pr...

Several Integrability Formulas of Special Functions

In this article, we give several integrability formulas of special and composite functions including trigonometric function, inverse trigonometric function, hyperbolic function and logarithmic function. The notation and terminology used here are introduced in the following papers:

Apostol-euler Polynomials of Higher Order and Gaussian Hypergeometric Functions

The purpose of this paper is to give analogous definitions of Apostol type (see T. M. Apostol [Pacific J. Math. 1 (1951), 161-167]) for the so-called Apostol-Euler numbers and polynomials of higher order. We establish their elementary properties, obtain several explicit formulas involving the Gaussian hypergeometric function and the Stirling numbers of the second kind, and deduce their special ...

Several Differentiation Formulas of Special Functions. Part III

The articles [13], [15], [16], [1], [4], [10], [11], [17], [5], [14], [12], [2], [6], [9], [7], [8], and [3] provide the terminology and notation for this paper. For simplicity, we follow the rules: x, r, a, b denote real numbers, n denotes a natural number, Z denotes an open subset of R, and f , f1, f2, f3 denote partial functions from R to R. One can prove the following propositions: (1) x2 Z...

Addition Theorems via Continued Fractions

We show connections between a special type of addition formulas and a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We apply our approach to derive special addition theorems for Bessel functions and confluent hypergeometric functions. We also derive several addition theorems for basic hypergeometric functions. Applications to the evaluat...