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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]

This document discusses recent advances in special functions and their applications across various scientific and engineering disciplines. It highlights the importance of special functions in fields such as mathematics, physics, engineering, statistics, astronomy, computer science, economics, chemistry, biology, medicine, materials science, and environmental science. The document provides summaries of 13 articles that explore topics such as Fibonacci polynomials, Appell and Horn functions, generalized hypergeometric functions, Lagrange-based hypergeometric Bernoulli polynomials, hypergeometric series, Humbert functions, integral formulas, twisted Catalan numbers, incomplete gamma function, existence theorems, finite sum identities, Euler polynomials, and q-generalized tangent-based Apostol-type Frobenius-Euler polynomials. These articles contribute to the understanding and application of special functions in various research realms. [Extracted from the article]

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]

In this article, we introduce three general double-series identities using Whipple transformations for terminating generalized hypergeometric 4 F 3 and 5 F 4 functions. Then, by employing the left-sided Riemann–Liouville fractional integral on these identities, we show the ability to derive additional identities of the same nature successively. These identities are used to derive transformation formulas between the Srivastava–Daoust double hypergeometric function (S–D function) and Kampé de Fériet's double hypergeometric function (KDF function) with equal arguments. We also demonstrate reduction formulas from the S–D function or KDF function to the generalized hypergeometric function p F q . Additionally, we provide general summation formulas for the p F q and S–D function (or KDF function) with specific arguments. We further highlight the connections between the results presented here and existing identities. [ABSTRACT FROM AUTHOR]

A very recent article delved into and expanded the four parametric linear Euler sums, revealing that two well-established subjects—Euler sums and series involving the zeta functions—display particular correlations. In this study, we present several closed forms of series involving zeta functions by using formulas for series associated with the zeta functions detailed in the aforementioned paper. Another closed form of series involving Riemann zeta functions is provided by utilizing a known identity for a series of rational functions in the series index, expressed in terms of Gamma functions. Furthermore, we demonstrate a myriad of applications and relationships of series involving the zeta functions and the extended parametric linear Euler sums. These include connections with Wallis's infinite product formula for π , Mathieu series, Mellin transforms, determinants of Laplacians, certain integrals expressed in terms of Euler sums, representations and evaluations of some integrals, and certain parametric Euler sum identities. The use of Mathematica for various approximation values and certain integral formulas is elaborated upon. Symmetry naturally occurs in Euler sums. [ABSTRACT FROM AUTHOR]

The Gould–Hopper–Laguerre–Sheffer matrix polynomials were initially studied using operational methods, but in this paper, we investigate them using matrix techniques. By leveraging properties of Pascal functionals and Wronskian matrices, we derive several identities for these polynomials, including recurrence relations. It is highlighted that these identities, acquired via matrix techniques, are distinct from the ones obtained when using operational methods. [ABSTRACT FROM AUTHOR]

This paper introduces extensions H 4 , p and X 8 , p of Horn's double hypergeometric function H 4 and Exton's triple hypergeometric function X 8 , taking into account recent extensions of Euler's beta function, hypergeometric function, and confluent hypergeometric function. Among the numerous extended hypergeometric functions, the primary rationale for choosing H 4 and X 8 is their comparable extension type. Next, we present various integral representations of Euler and Laplace types, Mellin and inverse Mellin transforms, Laguerre polynomial representations, transformation formulae, and a recurrence relation for the extended functions. In particular, we provide a generating function for X 8 , p and several bounding inequalities for H 4 , p and X 8 , p . We explore the utilization of the H 4 , p function within a probability distribution. Most special functions, such as the generalized hypergeometric function, the Beta function, and the p-extended Beta integral, exhibit natural symmetry. [ABSTRACT FROM AUTHOR]