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New reduction formulas for the Humbert functions (the confluent Appell functions),andare given. [ABSTRACT FROM AUTHOR]

Brychkov [Brychkov YuA. Evaluation of some classes of definite and indefinite integrals. Integral Transforms Spec Funct. 2002;13:163–167] evaluated some interesting classes of definite and indefinite integrals involving various special functions and the logarithmic function. Motivated essentially by Brychkov's work [Brychkov YuA. Evaluation of some classes of definite and indefinite integrals. Integral Transforms Spec Funct. 2002;13:163–167], we aim at presenting several classes of definite integrals involving hypergeometric functions and the logarithmic functions, which are expressed explicitly in terms of Psi and generalized zeta functions. Some interesting special cases of our main formulas are given. Relevant connections of the results presented here with those earlier ones are also pointed out. [ABSTRACT FROM AUTHOR]

The main objective of this investigation is to derive a generalization of Kummer’s formula: which was proved independently by Ramanujan. The results presented here are obtained with the help of a known generalization of one of Kummer’s summation theorems for 2 F 1 , which was given recently by Lavoie et al. [14]. Several interesting identities closely related to Ramanujan’s formula follow as special cases of our main results. [ABSTRACT FROM AUTHOR]

It is well known that classical summation theorems such as of Gauss, Gauss second, Kummer and Bailey for the series 2 F 1 and Watson, Dixon and Whipple for the series 3 F 2 play an important role in the theory of hypergeometric and generalized hypergeometric series. Applications of these summation theorems are well known. The aim of this research paper is to provide the generalizations of the above-mentioned classical summation theorems in the most general case. [ABSTRACT FROM AUTHOR]