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In this paper, we have developed novel proofs of two known formulas for the Meijer integral transform and obtained non-trivial extensions of these formulas (in one of them, the Bessel function is replaced by the Coulomb wave function). Also we have derived a new formula for a series involving product of Whittaker functions and modified Bessel functions of the second kind. Our proofs are based on a group theoretical approach and involves the use of an intertwining operator for two representations of one matrix group. [ABSTRACT FROM AUTHOR]

Using a representation of the unimodular Lorentz group, we derive some relations between hyper Bessel–Clifford, Macdonald and Meijer functions. We obtained them as two additional theorems (continual and countable) for a functional defined on the above group and a pair of basis functions belonging to representation spaces. Introducing a hyper analogue of the known first and second Hankel–Clifford integral transforms and writing the continual addition theorem for a particular case, we obtain a simple formula for the sum of these transforms of Macdonald function. [ABSTRACT FROM AUTHOR]

We aim to introduce certain unified polynomials which can be specialized to yield a number of known polynomials and numbers, for example, Pell polynomials and numbers, Fibonacci polynomials and numbers, and -Fibonacci polynomials. Then we investigate the unified polynomials to present certain interesting and potentially useful properties such as explicit expressions, some recurrence relations, a differential equation, and several generating functions. Further, among a number of polynomials and numbers which can be specialized from the unified polynomials, we choose the Pell polynomials to record their corresponding properties. [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]

Computing the matrix elements of the linear operator, which transforms the spherical basis ofSO(3, 1)-representation space into the hyperbolic basis, we present an integral formula involving the product of two Legendre functions of the first kind expressed in terms of4F3-hypergeometric function. Also, using the general Mehler–Fock transform, we obtain another integral formula for the Legendre function of the first kind. A relevant connection of one of the results presented here with a known integral formula is also pointed out. [ABSTRACT FROM PUBLISHER]

Two interesting infinite product formulas were presented by Choiet al. [A generalization of Wilf's formula, Kodai Math. J. 26 (2003), pp. 44–48], who generalized an infinite product formula posed by Wilf as a problem inAmerican Mathematical Monthly[H.S. Wilf,Problem 10588, problems and solutions, Amer. Math. Monthly 104 (1997), p. 456]. Here, we aim mainly at establishing two infinite product formulas, expressed in terms of the Gamma function, whose special cases contain the above generalized infinite product formulas. Our formulas presented here are themselves interesting and their another advantage is the production, easily, of various known or (presumably) new formulas as their special cases. [ABSTRACT FROM AUTHOR]

B.C. Carlson [Some extensions of Lardner’s relations between 0 F 3 and Bessel functions, SIAM J. Math. Anal. 1 (1970), pp. 232–242] presented several connections between Bessel functions and the generalized hypergeometric functions , which extended some earlier results of T.J. Lardner [Relations between 0 F 3 and Bessel functions, SIAM Rev. 11 (1969), pp. 69–72], being the set of nonpositive integers. In our investigation here, by simply splitting Lauricella’s triple hypergeometric function into eight parts, we prove some relations between and the Srivastava function . Furthermore, by suitably specializing the main results presented in this paper, we deduce a number of serendipitous relations between the functions 0 F 1 and 0 F 3, 1 F 1 and 2 F 3, and between a confluent Lauricella function and the Srivastava function . We also derive a system of partial differential equations satisfied by the Srivastava function . [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]

Motivated essentially by their potential for applications in a wide range of mathematical and physical problems, the log-sine integrals have been evaluated, in the existing literature on the subject, in many different ways. The main object of this paper is to present explicit evaluations of some families of log-sine integrals by making use of a familiar algorithm, and also those of log-cosine and other related integrals by using some known integral formulas expressed in terms of the Gamma functions. [ABSTRACT FROM AUTHOR]

Since the time of Euler, the so-called Euler-Mascheroni constant γ has been involved in a wide variety of important mathematical formulas and results. Among these formulas and results, a considerably large number of integral representations of γ have been developed. Here we aim at investigating several further integral expressions for the Euler-Mascheroni constant γ. [ABSTRACT FROM AUTHOR]

A class of definite integrals are evaluated using the Cauchy residue theorem. These evaluations are then used to obtain closed-form expressions for several infinite series associated with the Riemann zeta function. A number of interesting (known or new) special cases and consequences of the main results are also considered. [ABSTRACT FROM AUTHOR]