TY - JOUR AU - Agarwal, Ritu AU - Choi, Junesang AU - Kumar, Naveen AU - Parmar, Rakesh K. TI - EXTENSIONS OF MULTIPLE LAURICELLA AND HUMBERT'S CONFLUENT HYPERGEOMETRIC FUNCTIONS THROUGH A HIGHLY GENERALIZED POCHHAMMER SYMBOL AND THEIR RELATED PROPERTIES JF - BULLETIN OF THE KOREAN MATHEMATICAL SOCIETY J2 - B KOREAN MATH SOC VL - 60 PY - 2023 IS - 3 SP - 575 EP - 591 PG - 17 SN - 1015-8634 DO - 10.4134/BKMS.b210652 UR - https://m2.mtmt.hu/api/publication/34302987 ID - 34302987 AB - Motivated by several generalizations of the Pochhammer symbol and their associated families of hypergeometric functions and hy-pergeometric polynomials, by choosing to use a very generalized Pochham-mer symbol, we aim to introduce certain extensions of the generalized Lauricella function F(n) A and the Humbert's confluent hypergeometric function & psi;(n)of n variables with, as their respective particular cases, the second Appell hypergeometric function F2 and the generalized Hum-bert's confluent hypergeometric functions & psi;2 and investigate their several properties including, for example, various integral representations, finite summation formulas with an s-fold sum and integral representations in-volving the Laguerre polynomials, the incomplete gamma functions, and the Bessel and modified Bessel functions. Also, pertinent links between the major identities discussed in this article and different (existing or novel) findings are revealed. LA - English DB - MTMT ER - TY - JOUR AU - Chopra, Purnima AU - Gupta, Mamta AU - Modi, Kanak TI - FRACTIONAL INTEGRATION AND DIFFERENTIATION OF THE (p, q)-EXTENDED MODIFIED BESSEL FUNCTION OF THE SECOND KIND AND INTEGRAL TRANSFORMS JF - COMMUNICATIONS OF THE KOREAN MATHEMATICAL SOCIETY J2 - COMMUNICATIONS OF THE KOREAN MATHEMATICAL SOCIETY VL - 38 PY - 2023 IS - 3 SP - 755 EP - 772 PG - 18 SN - 1225-1763 DO - 10.4134/CKMS.c220132 UR - https://m2.mtmt.hu/api/publication/34621900 ID - 34621900 LA - English DB - MTMT ER - TY - JOUR AU - Ali, Musharraf AU - Ghayasuddin, Mohd AU - Paris, Richard Bruce TI - Extensions of beta and related functions JF - JOURNAL OF ANALYSIS J2 - J ANAL VL - 30 PY - 2022 IS - 2 SP - 717 EP - 729 PG - 13 SN - 0971-3611 DO - 10.1007/s41478-021-00363-0 UR - https://m2.mtmt.hu/api/publication/33331452 ID - 33331452 AB - In this paper, we introduce and investigate a new extension of the beta function by means of an integral operator involving a product of Bessel-Struve kernel functions. We also define a new extension of the well-known beta distribution, the Gauss hypergeometric function and the confluent hypergeometric function in terms of our extended beta function. In addition, some useful properties of these extended functions are also indicated in a systematic way. LA - English DB - MTMT ER - TY - JOUR AU - Chopra, Purnima AU - Gupta, Mamta AU - Modi, Kanak TI - CERTAIN IMAGE FORMULAS OF (p;nu){EXTENDED GAUSS' HYPERGEOMETRIC FUNCTION AND RELATED JACOBI TRANSFORMS JF - COMMUNICATIONS OF THE KOREAN MATHEMATICAL SOCIETY J2 - COMMUNICATIONS OF THE KOREAN MATHEMATICAL SOCIETY VL - 37 PY - 2022 IS - 4 SP - 1055 EP - 1072 PG - 18 SN - 1225-1763 DO - 10.4134/CKMS.c210344 UR - https://m2.mtmt.hu/api/publication/33911944 ID - 33911944 AB - Our aim is to establish certain image formulas of the (p,nu){extended Gauss' hypergeometric function F-p,F-nu (a; b; c; z) by using Saigo's hypergeometric fractional calculus (integral and differential) operators. Corresponding assertions for the classical Riemann-Liouville(R-L) and Erdelyi-Kober(E-K) fractional integral and differential operators are deduced. All the results are represented in terms of the Hadamard product of the (p,nu){extended Gauss's hypergeometric function F-p,F-nu (a; b; c; z) and Fox-Wright function r Psi s(z). We also established Jacobi and its particular assertions for the Gegenbauer and Legendre transforms of the (p,nu){extended Gauss' hypergeometric function F-p,F-nu (a; b; c; z). LA - English DB - MTMT ER - TY - JOUR AU - Dixit, Atul AU - Kesarwani, Aashita AU - Kumar, Rahul TI - Explicit transformations of certain Lambert series JF - RESEARCH IN THE MATHEMATICAL SCIENCES J2 - RES MATH SCI VL - 9 PY - 2022 IS - 2 PG - 54 SN - 2522-0144 DO - 10.1007/s40687-022-00331-5 UR - https://m2.mtmt.hu/api/publication/33331450 ID - 33331450 AB - An exact transformation, which we call the master identity, is obtained for the first time for the series Sigma(infinity)(n=1) sigma(a)(n)e(-ny) for a is an element of C and Re(y) > 0. New modular-type transformations when a is a nonzero even integer are obtained as its special cases. The precise obstruction to modularity is explicitly seen in these transformations. These include a novel companion to Ramanujan's famous formula for sigma (2m + 1). The Wigert-Bellman identity arising from the a = 0 case of the master identity is derived too. When a is an odd integer, the well-known modular transformations of the Eisenstein series on SL2 (Z), that of the Dedekind eta function as well as Ramanujan's formula for sigma (2m + 1) are derived from the master identity. The latter identity itself is derived using Guinand's version of the VoronoT summation formula and an integral evaluation of N. S. Koshliakov involving a generalization of the modified Bessel function K-v(z). Koshliakov's integral evaluation is proved for the first time. It is then generalized using a well-known kernel of Watson to obtain an interesting two-variable generalization of the modified Bessel function. This generalization allows us to obtain a new modular-type transformation involving the sums-of-squares function r(k)(n). Some results on functions self-reciprocal in the Watson kernel are also obtained. LA - English DB - MTMT ER - TY - JOUR AU - Gupta, Mamta AU - Modi, Kanak AU - Solanki, N. S. AU - Ali, Shoukat TI - Fractional integration and differentiation of the (p, q)- extended tau-hypergeometric function and related Jacobi transforms JF - JOURNAL OF ANALYSIS J2 - J ANAL VL - 30 PY - 2022 IS - 4 SP - 1817 EP - 1833 PG - 17 SN - 0971-3611 DO - 10.1007/s41478-022-00433-x UR - https://m2.mtmt.hu/api/publication/33331448 ID - 33331448 AB - The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee. In present paper, our aim is to the study and develop the compositions formulas of the generalized fractional calculus operators to obtain a number of key results for the (p, q)-extended by tau-hypergeometric function a R-p,q(tau) (ab; c; z) involving Saigo hypergeometric fractional integral and differential I operators in terms of the Hadamard product of the (p, q)-extended tau-hypergeometric function R-p,q(tau) (a, b; c; z) and Fox-Wright function (p)Psi(q)(z). Corresponding special cases results are obtained as particular choices of parameters reduces to the classical Riemann-Liouville and Erdelyi-Kober fractional integral and differential operators. We also established Jacobi and its particular assertions for the Gegenbauer and Legendre transforms of the (p, q)-extended r-hypergeometric function R-p,q(tau)(a, b; c; z). LA - English DB - MTMT ER - TY - JOUR AU - Bakhet, Ahmed AU - He, Fuli AU - Yu, Mimi TI - On the matrix version of extended Bessel functions and its application to matrix differential equations JF - LINEAR AND MULTILINEAR ALGEBRA J2 - LINEAR MULTILINEAR A PY - 2021 PG - 20 SN - 0308-1087 DO - 10.1080/03081087.2021.1923629 UR - https://m2.mtmt.hu/api/publication/32361702 ID - 32361702 AB - In this paper, we focus on the extensions of the Bessel matrix function and the modified Bessel matrix function. We first introduce the extended Bessel matrix function and the extended modified Bessel matrix function of the first kind by using the extended Beta matrix function. Then we establish the integral representations, differentiation formula, and hypergeometric representation of such functions. Finally, as an application, we study a kind of second-order matrix differential equations. We prove that the extended modified Bessel matrix function is a particular solution to this kind of differential equations. LA - English DB - MTMT ER - TY - JOUR AU - Habenom, Haile AU - Oli, Abdi AU - Suthar, D. L. TI - (p, q)-Extended Struve Function: Fractional Integrations and Application to Fractional Kinetic Equations JF - JOURNAL OF MATHEMATICS (HINDAWI) J2 - J MATH (HINDAWI) VL - 2021 PY - 2021 PG - 10 SN - 2314-4629 DO - 10.1155/2021/5536817 UR - https://m2.mtmt.hu/api/publication/32361703 ID - 32361703 AB - In this paper, the generalized fractional integral operators involving Appell's function F-3(center dot) in the kernel due to Marichev-Saigo-Maeda are applied to the (p, q)-extended Struve function. The results are stated in terms of Hadamard product of the Fox-Wright function (r)Psi(s) (z) and the (p, q)-extended Gauss hypergeometric function. A few of the special cases (Saigo integral operators) of our key findings are also reported in the corollaries. In addition, the solutions of a generalized fractional kinetic equation employing the concept of Laplace transform are also obtained and examined as an implementation of the (p, q)-extended Struve function. Technique and findings can be implemented and applied to a number of similar fractional problems in applied mathematics and physics. LA - English DB - MTMT ER - TY - JOUR AU - Parmar, Rakesh K. AU - Agarwal, Ritu AU - Kumar, Naveen AU - Purohit, S. D. TI - Extended elliptic-type integrals with associated properties and Turan-type inequalities JF - ADVANCES IN DIFFERENCE EQUATIONS J2 - ADV DIFFER EQU-NY VL - 2021 PY - 2021 IS - 1 PG - 16 SN - 1687-1839 DO - 10.1186/s13662-021-03536-0 UR - https://m2.mtmt.hu/api/publication/32301125 ID - 32301125 AB - Our aim is to study and investigate the family of (p, q)-extended (incomplete and complete) elliptic-type integrals for which the usual properties and representations of various known results of the (classical) elliptic integrals are extended in a simple manner. This family of elliptic-type integrals involves a number of special cases and has a connection with (p, q)-extended Gauss' hypergeometric function and (p, q)-extended Appell's double hypergeometric function F-1. Turan-type inequalities including log-convexity properties are proved for these (p, q)-extended complete elliptic-type integrals. Further, we establish various Mellin transform formulas and obtain certain infinite series representations containing Laguerre polynomials. We also obtain some relationship between these (p, q)-extended elliptic-type integrals and Meijer G-function of two variables. Moreover, we obtain several connections with (p, q)-extended beta function as special values and deduce numerous differential and integral formulas. In conclusion, we introduce (p, q)-extension of the Epstein-Hubbell (E-H) elliptic-type integral. LA - English DB - MTMT ER - TY - JOUR AU - Srivastava, Hari M. AU - AbuJarad, Eman S. A. AU - Jarad, Fahd AU - Srivastava, Gautam AU - AbuJarad, Mohammed H. A. TI - The Marichev-Saigo-Maeda Fractional-Calculus Operators Involving the (p,q)-Extended Bessel and Bessel-Wright Functions JF - FRACTAL AND FRACTIONAL J2 - FRACTAL FRACT VL - 5 PY - 2021 IS - 4 PG - 15 SN - 2504-3110 DO - 10.3390/fractalfract5040210 UR - https://m2.mtmt.hu/api/publication/33331451 ID - 33331451 AB - The goal of this article is to establish several new formulas and new results related to the Marichev-Saigo-Maeda fractional integral and fractional derivative operators which are applied on the (p,q)-extended Bessel function. The results are expressed as the Hadamard product of the (p,q)-extended Gauss hypergeometric function Fp,q and the Fox-Wright function r psi s(z). Some special cases of our main results are considered. Furthermore, the (p,q)-extended Bessel-Wright function is introduced. Finally, a variety of formulas for the Marichev-Saigo-Maeda fractional integral and derivative operators involving the (p,q)-extended Bessel-Wright function is established. LA - English DB - MTMT ER - TY - JOUR AU - Parmar, Rakesh K. AU - Pogany, Tibor TI - ON MATHIEU-TYPE SERIES FOR THE UNIFIED GAUSSIAN HYPERGEOMETRIC FUNCTIONS JF - APPLICABLE ANALYSIS AND DISCRETE MATHEMATICS J2 - APPL ANAL DISCR MATH VL - 14 PY - 2020 IS - 1 SP - 138 EP - 149 PG - 12 SN - 1452-8630 DO - 10.2298/AADM190525014P UR - https://m2.mtmt.hu/api/publication/31548211 ID - 31548211 AB - The main purpose of this paper is to present closed integral form expressions for the Mathieu-type alpha-series and for the associated alternating versions whose terms contain a generalized p-extended Gauss' hypergeometric function. Related bounding inequalities for the p-generalized Mathieu-type series are also obtained. Finally, a set of various (known or new) special cases and consequences of the results earned are presented. LA - English DB - MTMT ER - TY - JOUR AU - Parmar, Rakesh K. AU - Pogany, Tibor TI - ON (p, q)-EXTENSION OF FURTHER MEMBERS OF BESSEL-STRUVE FUNCTIONS CLASS JF - MISKOLC MATHEMATICAL NOTES J2 - MISKOLC MATH NOTES VL - 20 PY - 2019 IS - 1 SP - 451 EP - 463 PG - 13 SN - 1787-2405 DO - 10.18514/MMN.2019.2608 UR - https://m2.mtmt.hu/api/publication/31548213 ID - 31548213 AB - In [10] (p,q)-extensions of the modified Bessel and the modified Struve functions of the first kind are presented. This article companion to [10] contains the (p, q)-extension of modified Struve function of the second kind M-v(,p,q) and the Bessel-Struve kernel function S-v,S-p,S-q. Systematic investigation of its properties, among integral representation, Mellin transform, Laguerre polynomial representation for both introduced special functions, while additional differential-difference equation, log-convexity property and Turan-type inequalities are realized for the latter. LA - English DB - MTMT ER - TY - JOUR AU - Choi, Junesang AU - Parmar, Rakesh K. TI - FRACTIONAL INTEGRATION AND DIFFERENTIATION OF THE (p, q)-EXTENDED BESSEL FUNCTION JF - BULLETIN OF THE KOREAN MATHEMATICAL SOCIETY J2 - B KOREAN MATH SOC VL - 55 PY - 2018 IS - 2 SP - 599 EP - 610 PG - 12 SN - 1015-8634 DO - 10.4134/BKMS.b170193 UR - https://m2.mtmt.hu/api/publication/31548209 ID - 31548209 AB - We aim to present some formulas for Saigo hypergeometric fractional integral and differential operators involving (p, q)-extended Bessel function J(v, p, q)(z), which are expressed in terms of Hadamard product of the (p, q) -extended Gauss hypergeometric function and the Fox-Wright function p Psi q (Z). A number of interesting special cases of our main results are also considered. Further, it is emphasized that the results presented here, which are seemingly complicated series, can reveal their involved properties via those of the two known functions in their respective Hadamard product. LA - English DB - MTMT ER - TY - JOUR AU - Dixit, Atul AU - Kesarwani, Aashita AU - Moll, Victor H. TI - A generalized modified Bessel function and a higher level analogue of the theta transformation formula JF - JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS J2 - J MATH ANAL APPL VL - 459 PY - 2018 IS - 1 SP - 385 EP - 418 PG - 34 SN - 0022-247X DO - 10.1016/j.jmaa.2017.10.050 UR - https://m2.mtmt.hu/api/publication/31548208 ID - 31548208 AB - A new generalization of the modified Bessel function of the second kind K-z(x) is studied. Elegant series and integral representations, a differential-difference equation and asymptotic expansions are obtained for it thereby anticipating a rich theory that it may possess. The motivation behind introducing this generalization is to have a function which gives a new pair of functions reciprocal in the Koshliakov kernel cos (pi z) M-2z (4 root x) - sin (pi z) J(2z) (4 root x) and which subsumes the self-reciprocal pair involving K-z(x). Its application towards finding modular-type transformations of the form F(z, w, alpha) = F(z,iw,beta), where alpha beta = 1, is given. As an example, we obtain a beautiful generalization of a famous formula of Ramanujan and Guinand equivalent to the functional equation of a non-holomorphic Eisenstein series on SL2(Z). This generalization can be considered as a higher level analogue of the general theta transformation formula. We then use it to evaluate an integral involving the Riemann Xi-function and consisting of a sum of products of two confluent hypergeometric functions. (C) 2017 Elsevier Inc. All rights reserved. LA - English DB - MTMT ER - TY - JOUR AU - Rakesh, Kumar Parmar AU - Pogany, Tibor AU - Ram, Kishore Saxena TI - On properties and applications of (p,q)-extended τ-hypergeometric functions JF - COMPTES RENDUS MATHEMATIQUE J2 - CR MATH VL - 356 PY - 2018 IS - 3 SP - 278 EP - 282 PG - 5 SN - 1631-073X DO - 10.1016/j.crma.2017.12.014 UR - https://m2.mtmt.hu/api/publication/3339542 ID - 3339542 LA - English DB - MTMT ER - TY - CHAP AU - Baricz, Arpad AU - Masirevic, Dragana Jankov AU - Pogany, Tibor K. TI - Series of Bessel and Kummer-Type Functions Preface T2 - SERIES OF BESSEL AND KUMMER-TYPE FUNCTIONS PB - Springer Netherlands CY - Cham T3 - Lecture Notes in Mathematics, ISSN 0075-8434 PY - 2017 SP - VII EP - + PG - 17 UR - https://m2.mtmt.hu/api/publication/31539422 ID - 31539422 LA - English DB - MTMT ER -