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The study of probability distributions for random variables and their algebraic combinations has been a central focus driving the advancement of probability and statistics. Since the 1920s, the challenge of calculating the probability distributions of sums, differences, products, and quotients of independent random variables have drawn the attention of numerous statisticians and mathematicians who studied the algebraic properties and relationships of random variables. Statistical distributions are highly helpful in data science and machine learning, as they provide a range of possible values for the variables, aiding in the development of a deeper understanding of the underlying problem. In this paper, we have presented a new probability distribution based on the $\hat{I}$-function. Also, we have discussed the applications of the $\hat{I}$ function, particularly in deriving the distributions of product and the quotient involving two independent $\hat{I}$ function variates. Additionally, it has been shown that both the product and quotient of two independent $\hat{I}$-function variates also follow the $\hat{I}$-function distribution. Furthermore, the new distribution, known as the $\hat{I}$-function distribution, includes several well-known classical distributions such as the gamma, beta, exponential, normal H-function, and G-function distributions, among others, as special cases. Therefore, the $\hat{I}$-function distribution can be considered a characterization or generalization of the above-mentioned distributions., Comment: 14 pages

The objective of this short note is to provide two closed-form evaluations for the generalized hypergeometric function $_4F_3$ of the argument $\frac1{16}$. This is achieved by means of separating a generalized hypergeometric function $_3F_2$ into even and odd components, together with the use of two known results for $_3F_2(\pm\frac14)$ available in the literature. As an application, we obtain an interesting infinite-sum representation for the number $\pi^2$. Certain connections with the work of Ramanujan and other authors are discussed, involving other special functions and binomial sums of different kinds., Comment: Updated version with new references

We extend the Reed Dawson identity for Knuth's old sum with a complex parameter, and we offer two separate hypergeometric series-based proofs of this generalization, and we apply this generalization to introduce binomial-harmonic sum identities. We also provide another ${}_{2}F_{1}(2)$-generalization of the Reed Dawson identity involving a free parameter. We then apply Fourier-Legendre theory to obtain an identity involving odd harmonic numbers that resembles the formula for Knuth's old sum, and the modified Abel lemma on summation by parts is also applied., Comment: 12 pages

The aim of this paper is to provide a new class of series identities in the form of four general results. The results are established with the help of generalizatons of the classical Kummer's summation theorem obtained earlier by Rakha and Rathie. Results obtained earlier by Srivastava, Bailey and Rathie et al. follow special cases of our main findings., Comment: 5 pages

In this note, we aim to provide generalizations of (i) Knuth's old sum (or Reed Dawson identity) and (ii) Riordan's identity using a hypergeometric series approach., Comment: 5 pages

The aim of this research is to provide thirty-two interesting summation formulas for the Kamp\'e de F\'eriet function in general forms, which are given in sixteen theorems. The results are established with the help of the identities in Liu and Wang \cite{Li-Wa} and generalizations of Kummer's summation theorem, Gauss' second summation theorem and Bailey's summation theorem obtained earlier by Rakha and Rathie \cite{Ra-Ra}. Some special cases and relevant connections of the results presented here with those involving certain known identities are also indicated., Comment: 18 pages; submitted for publication

The main objective of this research note is to provide an identity for the H-function, which generalizes two identities involving H-function obtained earlier by Rathie and Rathie et al., Comment: 3 pages

Using generalized hypergeometric functions to perform symbolic manipulation of equations is of great importance to pure and applied scientists. There are in the literature a great number of identities for the Meijer-G function. On the other hand, when more complex expressions arise, the latter function is not capable of representing them. The H-function is an alternative to overcome this issue, as it is a generalization of the Meijer-G function. In the present paper, a new identity for the H-function is derived. In short, this result enables one to split a particular H-function into the sum of two other H-functions. The new relation in addition to an old result are applied to the summation of hypergeometric series. Finally, some relations between H-functions and elementary functions are built, Comment: 11 pages

The aim of this short note is to provide a very simple proof for obtaining the fundamental two-term transformation for the series ${}_3F_2(1)$ due to Thomae., Comment: 4 pages

In a recent paper, Chu and Zhou [Advances in Combinatorics, I.S. Kotsireas and E.V. Zima(eds.), 139-159 (2013)] established in all 40 closed formulae for terminating Watson-like hypergeometric $_{3}F_2$- series by investigating through Gould and Hsu's fundamental pair of inverse series relations, the dual relations of Dougall's formula for the very well - poised $_{5}F_4$ - series. The aim of this short note is just to point out that out of 40 results, 33 results have already been discovered in 1992 by Lavoie, et al., Comment: 2 pages

The aim of this note is to establish an interesting hypergeometric generating relation contiguous to that of Exton by a short method., Comment: 4 pages

Very recently a new series representation of Humbert's double hypergeometric series $\Phi_3$ in series of Gauss's $_2F_1$ function was given by one of us. The aim of this short research note is to provide an alternative proof of the result. A few interesting special cases are also given., Comment: 6 pages

Motivated by the new Laplace transforms for the Kummer's confluent hypergeometric functions $_1F_1$ obtained recently by Kim et al. [Math $\&$ Comput. Modelling, 55 (2012), pp. 1068--1071], the authors aim is to establish so far unknown Laplace transforms of rather general case of generalized hypergeometric functions $_2F_2(x)$ and $_3F_3(x)$ by employing extensions of classical summation theorems for the series $_2F_1$ and $_3F_2$ obtained recently by Kim et al. [Int. J. Math. Math. Sci., 309503, 26 pages, 2010]. Certain known results obtained earlier by Kim et al. follow cases of our main findings., Comment: 8 pages; 0 figure

By utilizing a two-dimensional extension of a very general series transform given by Bailey, Exton [Indian J. pure appl. Math. 24 (6) (1993), 401-408] deduced a very general double generating relation of a product of a pair of Laguerre polynomials and obtained a number of useful relations with elementary functions, Bessel functions, Hermite polynomials and single series expansions of pairs of Laguerre polynomials. Unfortunately, some of the results given by Exon contain errors and thus this is the aim of this short note to provide the corrected form of these results.

A simple proof of a new summation formula for a terminating r+3Fr+2(1) hypergeometric series, representing an extension of Saalschutz's formula for a 3F2(1) series, is given for the case of r pairs of numeratorial and denominatorial parameters differing by positive integers. Two applications of this extended summation theorem are discussed. The first application extends two identities given by Ramanujan and the second, which also employs a similar extension of the Vandermonde-Chu summation theorem for the 2F1 series, extends certain reduction formulas for the Kamp?e de F?eriet function of two variables given by Exton and Cvijovi?c and Miller., Comment: 12 pages

In this paper, we aim to obtain a representation of Humbert's hy- pergeometric function in a series of Gauss's function 2F1. A few interesting results have also been deduced as special case of our main findings., Comment: 4 pages

The aim of this research paper is to demonstrate how one can obtain eleven new and interesting hypergeometric identities (in the form of a single result) from the old ones by mainly applying the well known beta integral method which was used successfully and systematically by Krattenthaler and Rao in their well known, very interesting research papers. The results are derived with the help of generalization of a quadratic transformation formula due to Kummer very recently obtained by Kim, et al. . Several identities including one obtained earlier by Krattenthaler and Rao follow special cases of our main findings. The results established in this paper are simple, interesting, easily established and may be potentially useful., Comment: 8 pages

We obtain two new Thomae-type transformations for hypergeometric series with r pairs of numeratorial and denominatorial parameters differing by positive integers. This is achieved by application of the so-called Beta integral method developed by Krattenthaler and Rao [Symposium on Symmetries in Science (ed. B. Gruber), Kluwer (2004)] to two recently obtained Euler-type transformations. Some special cases are given., Comment: 6 pages

In this short research note, we aim to establish an interesting extension of a summation due to Ramanujan.The result is derived with the help of an extension of Gauss's summation theorem available in the literature.

During the course of verifying the results of Ramanujan on hypergeometric series, Berndt in his notebooks, Part II mentioned corrected forms of two of the Ramanujan's results. The aim of this short research note is to point out that one of the results obtained by Ramanujan is correct (and not of Berndt's result) and the second result (which is slightly differ from Ramanujan's result and Berndt's result) is given here in corrected form.

In general, while obtaining the probability density function of sums and products of shifted random variables, ordinary analytical methods such as Fourier and Mellin transforms tend to provide integrals which cannot be expressed in terms of ordinary Meijer G and H functions. This way, the need of defining new functions which easily enable one to write such integrals in a closed-form is inherent to the development of this area of statistical sciences. By generalizing the Mellin transform which defines the H function, a new function is established. A direct application of the so-called $\widehat{I}$ is discussed while developing the probability density function of the sum and the product of shifted generalized gamma random variables. Important special cases of the $\widehat{I}$ and their applications in science are also discussed in order to show the applicability of the function hereby defined.

In our present investigation we propose to study and develop the I-function of two variables analogous to the I-function of one variable introduced and studied by one of the authors[24]. The conditions for convergence, series representation, behavior for small values, elementary properties and some special cases for the I-functions of two variables are also discussed.

In this paper a natural generalization of the familiar H -function of Fox namely the I -function is proposed. Convergence conditions, various series representations, elementary properties and special cases for the I -function have also been given.

By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.