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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]

We present a novel proof, using group theory, for a Meijer transform formula. This proof reveals the formula as a specific case of a broader generalized result. The generalization is achieved through a linear operator that intertwines two representations of the connected component of the identity of the group SO (2 , 2) . Using this same approach, we derive a formula for the sum of three double integral transforms, where the kernels are represented by Bessel functions. It is particularly noteworthy that the group SO (2 , 2) is connected to symmetry in several significant ways, especially in mathematical physics and geometry. [ABSTRACT FROM AUTHOR]

We offer intriguing new insights into parameterized finite binomial sums, revealing elegant identities such as ∑ k = 0 , k ≠ n m + n (− 1) k n − k m + n k = (− 1) n m + n n (H m − H n) , where n , m are non-negative integers and H n is the harmonic number. These formulas beautifully capture the intricate relationship between harmonic numbers and binomial coefficients, providing a fresh and captivating perspective on combinatorial sums. [ABSTRACT FROM AUTHOR]

Considering a representation space for a group of unimodular diag (1 ,   − 1 ,   − 1) -matrices, we construct several bases whose elements are eigenfunctions of Casimir infinitesimal operators related to a reduction in the group to some one-parameter subgroups. Finding the kernels of base transformation integral operators in terms of special functions, we consider the compositions of some of these transformations. Since composition is a 'closed' operation on the set of base transformations, we obtain some integral relations for the special functions involved in the above kernels. [ABSTRACT FROM AUTHOR]

Numerous logarithmic integrals have been extensively documented in the literature. This paper presents an algorithmic evaluation of a specific class of these integrals. Our systematic approach, rooted in logarithmic principles, enables us to extend our findings to other cases within this family of integrals. Furthermore, we explore special cases derived from our main results, thereby enhancing the applicability and significance of our work for a wider audience of researchers. [ABSTRACT FROM AUTHOR]

A collection of functions organized according to their indexing based on non-negative integers is grouped by the common factor of fixed integer N. This grouping results in a summation of N series, each consisting of functions partitioned according to this modulo N rule. Notably, when N is equal to two, the functions in the series are divided into two subseries: one containing even-indexed functions and the other containing odd-indexed functions. This partitioning technique is widely utilized in the mathematical literature and finds applications in various contexts, such as in the theory of hypergeometric series. In this paper, we employ this partitioning technique to establish four distinct families of summation formulas for F 3 4 (1) hypergeometric series. Subsequently, we leverage these summation formulas to introduce eight categories of integral formulas. These integrals feature compositions of Beta function-type integrands and F 2 3 (x) hypergeometric functions. Additionally, we highlight that our primary summation formulas can be used to derive some well-known summation results. [ABSTRACT FROM AUTHOR]

The introduction of two-parameter (p , q) -calculus and Lie algebras in 1991 has spurred a wave of recent research into (p , q) -special polynomials, including (p , q) -Bernoulli, (p , q) -Euler, (p , q) -Genocchi and (p , q) -Frobenius–Euler polynomials. These investigations have been carried out by numerous researchers in order to uncover a wide range of identities associated with these polynomials and applications. In this article, we aim to introduce (p , q) -sine and (p , q) -cosine based λ-array type polynomials and derive numerous properties of these polynomials such as (p , q) -integral representations, (p , q) -partial derivative formulae and (p , q) -addition formulae. It is worth noting that the utilization of the (p , q) -polynomials introduced in this study, along with other (p , q) -polynomials, can lead to the derivation of various identities that differ from the ones presented here. [ABSTRACT FROM AUTHOR]