{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,3]],"date-time":"2026-08-03T19:37:19Z","timestamp":1785785839885,"version":"3.56.0"},"reference-count":42,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,4,18]],"date-time":"2021-04-18T00:00:00Z","timestamp":1618704000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the Pochhammer symbol, and hypergeometric functions, this work is devoted to the study of the k-analogue of Gauss hypergeometric functions by the Hadamard product. We give a definition of the Hadamard product of k-Gauss hypergeometric functions (HPkGHF) associated with the fourth numerator and two denominator parameters. In addition, convergence properties are derived from this function. We also discuss interesting properties such as derivative formulae, integral representations, and integral transforms including beta transform and Laplace transform. Furthermore, we investigate some contiguous function relations and differential equations connecting the HPkGHF. The current results are more general than previous ones. Moreover, the proposed results are useful in the theory of k-special functions where the hypergeometric function naturally occurs.<\/jats:p>","DOI":"10.3390\/sym13040714","type":"journal-article","created":{"date-parts":[[2021,4,19]],"date-time":"2021-04-19T21:59:49Z","timestamp":1618869589000},"page":"714","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Investigation of the k-Analogue of Gauss Hypergeometric Functions Constructed by the Hadamard Product"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4686-2143","authenticated-orcid":false,"given":"Mohamed","family":"Abdalla","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6815-2615","authenticated-orcid":false,"given":"Muajebah","family":"Hidan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Agarwal, P., Agarwal, R., and Ruzhansky, M. (2020). Special Functions and Analysis of Differential Equations, CRC Press. [1st ed.].","DOI":"10.1201\/9780429320026"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Akhmedova, V., and Akhmedov, E. (2019). Selected Special Functions for Fundamental Physics, Springer. SpringerBriefs in Physics.","DOI":"10.1007\/978-3-030-35089-5"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Seaborn, J. (1991). Hypergeometric Functions and Their Applications. Springer.","DOI":"10.1007\/978-1-4757-5443-8"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Sneddon, I. (1956). Special Functions of Mathematical Physics and Chemistry, Oliver and Boyd.","DOI":"10.1063\/1.3059825"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2601","DOI":"10.1007\/s40995-019-00756-8","article-title":"Some extensions of the Pochhammer Symbol and the associated hypergeometric functions","volume":"43","author":"Srivastava","year":"2019","journal-title":"Iran. J. Sci. Technol. Trans. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Srivastava, H., Tassaddiq, A., Rahman, G., Nisar, K., and Khan, I. (2019). A new extension of the t-Gauss hypergeometric function and its associated properties. Mathematics, 7.","DOI":"10.3390\/math7100996"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"585","DOI":"10.22199\/issn.0717-6279-2019-03-0037","article-title":"Note on extended hypergeometric function","volume":"38","author":"Jana","year":"2019","journal-title":"Proyecciones (Antofagasta)"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"70","DOI":"10.18311\/jims\/2020\/24874","article-title":"Some results on the extended hypergeometric function","volume":"87","author":"Jana","year":"2020","journal-title":"J. Indian Math. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"139","DOI":"10.18576\/amis\/120113","article-title":"A note on the new extended beta and Gauss hypergeometric functions","volume":"12","author":"Goswami","year":"2018","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"403","DOI":"10.4134\/CKMS.2015.30.4.403","article-title":"Extended hypergeometric functions of two and three variables","volume":"30","author":"Agarwal","year":"2015","journal-title":"Commun. Korean Math. Soc."},{"key":"ref_11","first-page":"4268361","article-title":"On the extended hypergeometric matrix functions and their applications for the derivatives of the extended Jacobi matrix polynomial","volume":"2020","author":"Fuli","year":"2020","journal-title":"Math. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"5535962","DOI":"10.1155\/2021\/5535962","article-title":"Further results on the (p; k) analogue of hypergeometric functions associated with fractional calculus operators","volume":"2021","author":"Hidan","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6058987","DOI":"10.1155\/2020\/6058987","article-title":"A note on the Appell hypergeometric matrix function F2","volume":"2020","author":"Hidan","year":"2020","journal-title":"Math. Probl. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/03081087.2018.1497585","article-title":"Special matrix functions: Characteristics, achievements and future directions","volume":"68","author":"Abdalla","year":"2020","journal-title":"Linear Multilinear Algebra"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1186\/s13662-020-02704-y","article-title":"Fractional operators for the Wright hypergeometric matrix functions","volume":"2020","author":"Abdalla","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1504\/IJCSM.2019.097642","article-title":"Further results on the generalized hypergeometric matrix functions","volume":"10","author":"Abdalla","year":"2019","journal-title":"Int. J. Comput. Sci. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1007\/s40995-017-0183-3","article-title":"Extended Gauss hypergeometric matrix functions","volume":"42","author":"Abdalla","year":"2018","journal-title":"Iran. J. Sci. Technol. Trans. Sci."},{"key":"ref_18","first-page":"179","article-title":"On hypergeometric functions and k- Pochhammer symbol","volume":"15","author":"Diaz","year":"2007","journal-title":"Divulg. Mat."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1155\/2014\/128787","article-title":"Solutions of k-hypergeometric differential equations","volume":"2014","author":"Mubeen","year":"2014","journal-title":"J. Appl. Math."},{"key":"ref_20","first-page":"410801","article-title":"Contiguous function relations for k-hypergeometric Functions","volume":"2014","author":"Mubeen","year":"2014","journal-title":"ISRN Math. Anal."},{"key":"ref_21","first-page":"203","article-title":"An integral representation of some k-hypergeometric functions","volume":"7","author":"Mubeen","year":"2012","journal-title":"Int. Math. Forum."},{"key":"ref_22","first-page":"41","article-title":"k-Analogue of Kummers first formula","volume":"3","author":"Mubeen","year":"2012","journal-title":"J. Inequal. Spec. Funct."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1936","DOI":"10.3934\/math.2020129","article-title":"On generalized k- fractional derivative operator","volume":"5","author":"Rahman","year":"2020","journal-title":"AIMS Math."},{"key":"ref_24","first-page":"637","article-title":"On several new contiguous function relations for k-hypergeometric function with two parameters","volume":"32","author":"Chinra","year":"2017","journal-title":"Commun. Korean Math. Soc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"119","DOI":"10.17706\/ijapm.2019.9.3.119-126","article-title":"Generating functions for k-hypergeometric functions","volume":"9","year":"2019","journal-title":"Int. J. Appl. Phys. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1186\/s13660-018-1717-8","article-title":"Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function","volume":"2018","author":"Nisar","year":"2018","journal-title":"J. Inequal. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Li, S., and Dong, Y. (2019). k-hypergeometric series solutions to one type of non-homogeneous k-hypergeometric equations. Symmetry, 11.","DOI":"10.3390\/sym11020262"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Kiryakova, V. (2020). Unified approach to fractional calculus images of special functions\u2014A survey. Mathematics, 8.","DOI":"10.3390\/math8122260"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Yilmazer, R., and Ali, K. (2020). Discrete fractional solutions to the k-hypergeometric differential equation. Math. Meth. Appl. Sci., 18.","DOI":"10.1002\/mma.6460"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/S0007-4497(01)01104-6","article-title":"The Hadamard product of hypergeometric series","volume":"126","author":"Sadykov","year":"2002","journal-title":"Bull. Sci. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"8596736","DOI":"10.1155\/2020\/8596736","article-title":"Certain generating relations involving the generalized multi-index Bessel\u2014Maitland and function","volume":"2020","author":"Jain","year":"2020","journal-title":"Math. Probl. Eng."},{"key":"ref_32","unstructured":"Rainville, E. (1960). Special Functions, The Macmillan."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"257","DOI":"10.2298\/AADM171017002M","article-title":"A Study Of generalized summation theorems for the series 2F1 with an applications to laplace transforms of convolution type integrals involving Kummer\u2019s functions 1F1","volume":"12","author":"Milovanovi","year":"2018","journal-title":"Appl. Anal. Discret. Math."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Koepf, W., Kim, I., and Rathie, A. (2019). On a new class of Laplace-type integrals involving generalized hypergeometric functions. Axioms, 8.","DOI":"10.3390\/axioms8030087"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3041","DOI":"10.1016\/j.cpc.2014.07.014","article-title":"Hypergeometric functions differential reduction (hyperdire): Mathematic Abased packages for differential reduction of generalized hypergeometric functions: FD and FS Horn-type hypergeometric functions of three variables","volume":"185","author":"Byteva","year":"2014","journal-title":"Comput. Phys. Commun."},{"key":"ref_36","first-page":"5510387","article-title":"On existence of sequences of weak solutions of fractional systems with Lipschitz nonlinearity","volume":"2021","author":"Guefaifia","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_37","first-page":"5544740","article-title":"On existence of multiplicity of weak solutions for a new class of nonlinear fractional boundary value systems via variational approach","volume":"2021","author":"KamKamache","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2236","DOI":"10.1016\/j.jmaa.2019.07.053","article-title":"Product of two hypergeometric functions with power arguments","volume":"479","author":"Kaiblinger","year":"2019","journal-title":"J. Math. Anal. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"4664","DOI":"10.3934\/math.2021274","article-title":"Blow up of solutions for a system of two singular nonlocal viscoelastic equations with damping, general source terms and a wide class of relaxation functions","volume":"6","author":"Boulaaras","year":"2021","journal-title":"AIMS Math."},{"key":"ref_40","first-page":"9948143","article-title":"On the Porous-Elastic System with Thermoelasticity of Type III and Distributed Delay: Well-Posedness and Stability","volume":"2021","author":"Ouchenane","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_41","first-page":"5577277","article-title":"On the system of coupled nonde generate Kirchhoff equations with distributed delay: Global existence and exponential decay","volume":"2021","author":"Choucha","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"110","DOI":"10.3906\/mat-2007-67","article-title":"On the generalization of k-fractional Hilfer-Katugampola derivative with Cauchy problem","volume":"45","author":"Naz","year":"2021","journal-title":"Turk. J. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/4\/714\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:49:29Z","timestamp":1760161769000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/4\/714"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,18]]},"references-count":42,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,4]]}},"alternative-id":["sym13040714"],"URL":"https:\/\/doi.org\/10.3390\/sym13040714","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,18]]}}}