@article{MTMT:34302987, title = {EXTENSIONS OF MULTIPLE LAURICELLA AND HUMBERT'S CONFLUENT HYPERGEOMETRIC FUNCTIONS THROUGH A HIGHLY GENERALIZED POCHHAMMER SYMBOL AND THEIR RELATED PROPERTIES}, url = {https://m2.mtmt.hu/api/publication/34302987}, author = {Agarwal, Ritu and Choi, Junesang and Kumar, Naveen and Parmar, Rakesh K.}, doi = {10.4134/BKMS.b210652}, journal-iso = {B KOREAN MATH SOC}, journal = {BULLETIN OF THE KOREAN MATHEMATICAL SOCIETY}, volume = {60}, unique-id = {34302987}, issn = {1015-8634}, abstract = {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.}, keywords = {Laguerre polynomials; Lauricella Functions; Appell functions; Generalized gamma functions; generalized Pochhammer symbol; generalized Gauss hypergeometric function; incomplete gamma functions; Bessel and modified Bessel functions; general-ized second Appell function in two variables; generalized multivariable Lauricella functions; generalized multivariable Humbert's confluent hypergeometric function}, year = {2023}, eissn = {1015-8634}, pages = {575-591} } @article{MTMT:34621900, title = {FRACTIONAL INTEGRATION AND DIFFERENTIATION OF THE (p, q)-EXTENDED MODIFIED BESSEL FUNCTION OF THE SECOND KIND AND INTEGRAL TRANSFORMS}, url = {https://m2.mtmt.hu/api/publication/34621900}, author = {Chopra, Purnima and Gupta, Mamta and Modi, Kanak}, doi = {10.4134/CKMS.c220132}, journal-iso = {COMMUNICATIONS OF THE KOREAN MATHEMATICAL SOCIETY}, journal = {COMMUNICATIONS OF THE KOREAN MATHEMATICAL SOCIETY}, volume = {38}, unique-id = {34621900}, issn = {1225-1763}, keywords = {fractional calculus operators; Extended beta function; (p, q)-extended Gauss hypergeometric function; F-p,F-q(a,b c;z); (p,q)-extended modified Bessel function of the second kind M-nu p,M-q(z)}, year = {2023}, pages = {755-772} } @article{MTMT:33331452, title = {Extensions of beta and related functions}, url = {https://m2.mtmt.hu/api/publication/33331452}, author = {Ali, Musharraf and Ghayasuddin, Mohd and Paris, Richard Bruce}, doi = {10.1007/s41478-021-00363-0}, journal-iso = {J ANAL}, journal = {JOURNAL OF ANALYSIS}, volume = {30}, unique-id = {33331452}, issn = {0971-3611}, abstract = {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.}, keywords = {Gauss hypergeometric function; Confluent hypergeometric function; beta function; Extended beta function; Extended Gauss hypergeometric function; Extended confluent hypergeometric function; Bessel-Struve kernel function; Extended beta distribution}, year = {2022}, eissn = {2367-2501}, pages = {717-729}, orcid-numbers = {Ali, Musharraf/0000-0001-9791-3217} } @article{MTMT:33911944, title = {CERTAIN IMAGE FORMULAS OF (p;nu){EXTENDED GAUSS' HYPERGEOMETRIC FUNCTION AND RELATED JACOBI TRANSFORMS}, url = {https://m2.mtmt.hu/api/publication/33911944}, author = {Chopra, Purnima and Gupta, Mamta and Modi, Kanak}, doi = {10.4134/CKMS.c210344}, journal-iso = {COMMUNICATIONS OF THE KOREAN MATHEMATICAL SOCIETY}, journal = {COMMUNICATIONS OF THE KOREAN MATHEMATICAL SOCIETY}, volume = {37}, unique-id = {33911944}, issn = {1225-1763}, abstract = {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).}, keywords = {Extended beta function; (p,nu)-extended Gauss hypergeometric function F-p,F-nu (a,b,c,z); fractional calculus operators.}, year = {2022}, pages = {1055-1072} } @article{MTMT:33331450, title = {Explicit transformations of certain Lambert series}, url = {https://m2.mtmt.hu/api/publication/33331450}, author = {Dixit, Atul and Kesarwani, Aashita and Kumar, Rahul}, doi = {10.1007/s40687-022-00331-5}, journal-iso = {RES MATH SCI}, journal = {RESEARCH IN THE MATHEMATICAL SCIENCES}, volume = {9}, unique-id = {33331450}, issn = {2522-0144}, abstract = {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.}, keywords = {Bessel functions; Watson kernel; Modular transformations; Ramanujan's formula for odd zeta values; Sums-of-squares function}, year = {2022}, eissn = {2197-9847} } @article{MTMT:33331448, title = {Fractional integration and differentiation of the (p, q)- extended tau-hypergeometric function and related Jacobi transforms}, url = {https://m2.mtmt.hu/api/publication/33331448}, author = {Gupta, Mamta and Modi, Kanak and Solanki, N. S. and Ali, Shoukat}, doi = {10.1007/s41478-022-00433-x}, journal-iso = {J ANAL}, journal = {JOURNAL OF ANALYSIS}, volume = {30}, unique-id = {33331448}, issn = {0971-3611}, abstract = {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).}, keywords = {(p, q)-extended hypergeometric function; fractional calculus operators; Legendre transforms; (p, q)-extended tau-hypergeometric function; Jacobi transforms; Gegenbauer transforms}, year = {2022}, eissn = {2367-2501}, pages = {1817-1833} } @article{MTMT:32361702, title = {On the matrix version of extended Bessel functions and its application to matrix differential equations}, url = {https://m2.mtmt.hu/api/publication/32361702}, author = {Bakhet, Ahmed and He, Fuli and Yu, Mimi}, doi = {10.1080/03081087.2021.1923629}, journal-iso = {LINEAR MULTILINEAR A}, journal = {LINEAR AND MULTILINEAR ALGEBRA}, unique-id = {32361702}, issn = {0308-1087}, abstract = {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.}, keywords = {integral representation; Bessel matrix function; differentiation formula; hypergeometric representation; matrix differential equation}, year = {2021}, eissn = {1563-5139} } @article{MTMT:32361703, title = {(p, q)-Extended Struve Function: Fractional Integrations and Application to Fractional Kinetic Equations}, url = {https://m2.mtmt.hu/api/publication/32361703}, author = {Habenom, Haile and Oli, Abdi and Suthar, D. L.}, doi = {10.1155/2021/5536817}, journal-iso = {J MATH (HINDAWI)}, journal = {JOURNAL OF MATHEMATICS (HINDAWI)}, volume = {2021}, unique-id = {32361703}, issn = {2314-4629}, abstract = {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.}, year = {2021}, eissn = {2314-4785}, orcid-numbers = {Oli, Abdi/0000-0001-5944-2442} } @article{MTMT:32301125, title = {Extended elliptic-type integrals with associated properties and Turan-type inequalities}, url = {https://m2.mtmt.hu/api/publication/32301125}, author = {Parmar, Rakesh K. and Agarwal, Ritu and Kumar, Naveen and Purohit, S. D.}, doi = {10.1186/s13662-021-03536-0}, journal-iso = {ADV DIFFER EQU-NY}, journal = {ADVANCES IN DIFFERENCE EQUATIONS}, volume = {2021}, unique-id = {32301125}, issn = {1687-1839}, abstract = {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.}, keywords = {Laguerre polynomials; Mellin transform; Turan-type inequalities; elliptic integrals; Extended beta function; Extended hypergeometric functions}, year = {2021}, eissn = {1687-1847} } @article{MTMT:33331451, title = {The Marichev-Saigo-Maeda Fractional-Calculus Operators Involving the (p,q)-Extended Bessel and Bessel-Wright Functions}, url = {https://m2.mtmt.hu/api/publication/33331451}, author = {Srivastava, Hari M. and AbuJarad, Eman S. A. and Jarad, Fahd and Srivastava, Gautam and AbuJarad, Mohammed H. A.}, doi = {10.3390/fractalfract5040210}, journal-iso = {FRACTAL FRACT}, journal = {FRACTAL AND FRACTIONAL}, volume = {5}, unique-id = {33331451}, abstract = {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.}, keywords = {Fox-Wright function; (p, q)-extended Bessel function; operators of fractional calculus; (p, q)-extensions of special functions; (p, q)-extended Gauss hypergeometric function; (p, q)-extended Bessel-Wright function; Marichev-Saigo-Maeda fractional integral and fractional derivative operators; Euler-Darboux partial differential equation}, year = {2021}, eissn = {2504-3110}, orcid-numbers = {Srivastava, Hari M./0000-0002-9277-8092; Srivastava, Gautam/0000-0001-9851-4103} } @article{MTMT:31548211, title = {ON MATHIEU-TYPE SERIES FOR THE UNIFIED GAUSSIAN HYPERGEOMETRIC FUNCTIONS}, url = {https://m2.mtmt.hu/api/publication/31548211}, author = {Parmar, Rakesh K. and Pogany, Tibor}, doi = {10.2298/AADM190525014P}, journal-iso = {APPL ANAL DISCR MATH}, journal = {APPLICABLE ANALYSIS AND DISCRETE MATHEMATICS}, volume = {14}, unique-id = {31548211}, issn = {1452-8630}, abstract = {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.}, keywords = {Integral representations; Mathieu-type series; Generalized p-extended Beta function; Generalized p-extended Gauss' hypergeometric function; Cahen formula; bounding inequality}, year = {2020}, eissn = {1452-8630}, pages = {138-149} } @article{MTMT:31548213, title = {ON (p, q)-EXTENSION OF FURTHER MEMBERS OF BESSEL-STRUVE FUNCTIONS CLASS}, url = {https://m2.mtmt.hu/api/publication/31548213}, author = {Parmar, Rakesh K. and Pogany, Tibor}, doi = {10.18514/MMN.2019.2608}, journal-iso = {MISKOLC MATH NOTES}, journal = {MISKOLC MATHEMATICAL NOTES}, volume = {20}, unique-id = {31548213}, issn = {1787-2405}, abstract = {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.}, keywords = {integral representation; log-convexity; Turan-type inequality; (p,q)-extended Beta function; (p,q)-extended Bessel and modified Bessel and Struve functions of the first and second kind; (p,q)-extended Bessel-Struve kernel function}, year = {2019}, eissn = {1787-2413}, pages = {451-463} } @article{MTMT:31548209, title = {FRACTIONAL INTEGRATION AND DIFFERENTIATION OF THE (p, q)-EXTENDED BESSEL FUNCTION}, url = {https://m2.mtmt.hu/api/publication/31548209}, author = {Choi, Junesang and Parmar, Rakesh K.}, doi = {10.4134/BKMS.b170193}, journal-iso = {B KOREAN MATH SOC}, journal = {BULLETIN OF THE KOREAN MATHEMATICAL SOCIETY}, volume = {55}, unique-id = {31548209}, issn = {1015-8634}, abstract = {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.}, keywords = {Fox-Wright function; hadamard product; (p, q)-extended Bessel function; (p, q)-extended hypergeometric function; fractional calculus operators}, year = {2018}, eissn = {1015-8634}, pages = {599-610} } @article{MTMT:31548208, title = {A generalized modified Bessel function and a higher level analogue of the theta transformation formula}, url = {https://m2.mtmt.hu/api/publication/31548208}, author = {Dixit, Atul and Kesarwani, Aashita and Moll, Victor H.}, doi = {10.1016/j.jmaa.2017.10.050}, journal-iso = {J MATH ANAL APPL}, journal = {JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, volume = {459}, unique-id = {31548208}, issn = {0022-247X}, abstract = {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.}, keywords = {Bessel functions; Asymptotic expansion; Riemann Xi-function; Theta transformation formula; Basset's formula; Ramanujan-Guinand formula}, year = {2018}, eissn = {1096-0813}, pages = {385-418} } @article{MTMT:3339542, title = {On properties and applications of (p,q)-extended τ-hypergeometric functions}, url = {https://m2.mtmt.hu/api/publication/3339542}, author = {Rakesh, Kumar Parmar and Pogany, Tibor and Ram, Kishore Saxena}, doi = {10.1016/j.crma.2017.12.014}, journal-iso = {CR MATH}, journal = {COMPTES RENDUS MATHEMATIQUE}, volume = {356}, unique-id = {3339542}, issn = {1631-073X}, year = {2018}, eissn = {1778-3569}, pages = {278-282} } @{MTMT:31539422, title = {Series of Bessel and Kummer-Type Functions Preface}, url = {https://m2.mtmt.hu/api/publication/31539422}, author = {Baricz, Arpad and Masirevic, Dragana Jankov and Pogany, Tibor K.}, booktitle = {SERIES OF BESSEL AND KUMMER-TYPE FUNCTIONS}, unique-id = {31539422}, year = {2017}, pages = {VII-+} }