On basic Horn hypergeometric functions \(\mathrm{H}_3\) and \(\mathrm{H}_4\). (English) Zbl 1486.33021
Summary: The purpose of this work is to demonstrate several interesting contiguous function relations and \(q\)-differential formulas for basic Horn hypergeometric functions \(\mathrm{H}_3\) and \(\mathrm{H}_4\). Some properties of our main results are also constructed.
MSC:
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
| 33D70 | Other basic hypergeometric functions and integrals in several variables |
| 05A30 | \(q\)-calculus and related topics |
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 11B83 | Special sequences and polynomials |
Keywords:
quantum calculus; \(q\)-basic Horn hypergeometric functions; contiguous relations; \(q\)-derivativesReferences:
| [1] | Acikgoz, M.; Araci, S.; Duran, U., New extensions of some known special polynomials under the theory of multiple q-calculus, Turk. J. Anal. Number Theory, 3, 128-139 (2015) · doi:10.12691/tjant-3-5-4 |
| [2] | Agarwal, R. P., Some relations between basic hypergeometric functions of two variables, Rend. Circ. Mat. Palermo (2), 3, 1-7 (1954) · Zbl 0055.30204 · doi:10.1007/BF02849368 |
| [3] | Agarwal, R. P., Contiguous relations for bilateral basic hypergeometric series, Int. J. Math. Sci., 3, 2, 375-388 (2004) · Zbl 1266.33017 |
| [4] | Andrews, G. E., On q-difference equations for certain well-poised basic hypergeometric series, Quart. J. Math., 19, 2, 433-447 (1968) · Zbl 0165.08202 · doi:10.1093/qmath/19.1.433 |
| [5] | Andrews, G. E.; Askey, R. A., Problems and Prospects for Basic Hypergeometric Functions, Theory and Applications of Special Functions (1975), San Diego: Academic Press, San Diego · Zbl 0342.33001 |
| [6] | Araci, S.; Acikgoz, M.; Park, K. H.; Jolany, H., On the unification of two families of multiple twisted type polynomials by using q-adic q-integral at \(q=-1\), Bull. Malays. Math. Sci. Soc., 37, 17-41 (2014) |
| [7] | Araci, S.; Duran, U.; Acikgoz, M., Symmetric identities involving q-Frobenius-Euler polynomials under Sym (4), Turk. J. Anal. Number Theory, 3, 90-93 (2015) · doi:10.12691/tjant-3-3-5 |
| [8] | Bagdasaryan, A.; Araci, S.; Acikgoz, M.; He, Y., Some new identities on the Apostol-Bernoulli polynomials of higher order derived from Bernoulli basis, J. Nonlinear Sci. Appl., 9, 2697-2704 (2016) · Zbl 1338.11025 · doi:10.22436/jnsa.009.05.66 |
| [9] | Bansal, M. K.; Choi, J., A note on pathway fractional integral formulas associated with the incomplete H-functions, Int. J. Appl. Comput. Math., 5, 5 (2019) · Zbl 1428.26010 · doi:10.1007/s40819-019-0718-8 |
| [10] | Bansal, M. K.; Kumar, D., On the integral operators pertaining to a family of incomplete I-functions, AIMS Math., 5, 2, 1247-1259 (2020) · Zbl 1485.33014 · doi:10.3934/math.2020085 |
| [11] | Bansal, M. K.; Kumar, D.; Jain, R., A study of Marichev-Saigo-Maeda fractional integral operators associated with S-generalized Gauss hypergeometric function, Kyungpook Math. J., 59, 3, 433-443 (2019) · Zbl 1434.33012 |
| [12] | Bansal, M. K.; Kumar, D.; Nisar, K. S.; Singh, J., Certain fractional calculus and integral transform results of incomplete ℵ-functions with applications, Math. Methods Appl. Sci., 43, 8, 5602-5614 (2020) · Zbl 1445.26002 · doi:10.1002/mma.6299 |
| [13] | Duran, U.; Acikgoz, M.; Araci, S., On a class of multiple q-Bernoulli, multiple q-Euler and multiple q-Genocchi polynomials of order α, Adv. Appl. Math. Sci., 15, 8, 229-254 (2016) · Zbl 1386.11042 |
| [14] | Duran, U.; Acikgoz, M.; Esi, A.; Araci, S., Some new symmetric identities involving q-Genocchi polynomials under \(S_4\), J. Math. Anal., 6, 5, 22-31 (2015) · Zbl 1362.33029 |
| [15] | Gasper, G.; Rahman, M., Basic Hypergeometric Series (2004), Cambridge: Cambridge University Press, Cambridge · Zbl 1129.33005 |
| [16] | Gupta, D., Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. III. Associated continuous dual q-Hahn polynomials, J. Comput. Appl. Math., 68, 1-2, 115-149 (1996) · Zbl 0878.33011 · doi:10.1016/0377-0427(95)00264-2 |
| [17] | Ismail, M. E.H., Classical and Quantum Orthogonal Polynomials in One Variable (2009), Cambridge: Cambridge University Press, Cambridge · Zbl 1172.42008 |
| [18] | Ismail, M. E.H.; Libis, C. A., Contiguous relations, basic hypergeometric functions, and orthogonal polynomials I, J. Math. Anal. Appl., 141, 2, 349-372 (1989) · Zbl 0681.33011 · doi:10.1016/0022-247X(89)90182-0 |
| [19] | Jackson, F. H., On basic double hypergeometric functions, Quart. J. Math., 13, 69-82 (1942) · Zbl 0060.19809 · doi:10.1093/qmath/os-13.1.69 |
| [20] | Jackson, F. H., On some formulae in partition theory, and bilateral basic hypergeometric series, J. Lond. Math. Soc., 24, 233-237 (1949) · Zbl 0034.34003 · doi:10.1112/jlms/s1-24.3.233 |
| [21] | Jain, V. K., Some transformations of basic hypergeometric series and their applications, Proc. Am. Math. Soc., 78, 375-384 (1980) · Zbl 0436.33003 · doi:10.1090/S0002-9939-1980-0553380-8 |
| [22] | Jain, V. K., Some transformations of basic hypergeometric functions. Part II, SIAM J. Math. Anal., 12, 957-961 (1981) · Zbl 0469.33003 · doi:10.1137/0512081 |
| [23] | Jain, V. K.; Vertna, A., Transformations between basic hypergeometric series on different bases and identities of Rogers-Ramanujan type, J. Math. Anal. Appl., 76, 230-269 (1980) · Zbl 0443.33004 · doi:10.1016/0022-247X(80)90076-1 |
| [24] | Mishra, B. P., On certain transformation formulae involving basic hypergeometric functions, J. Ramanujan Soc. Math. Math. Sci., 2, 2, 9-16 (2014) · Zbl 1445.33005 |
| [25] | Sahai, V.; Verma, A., nth-order q-derivatives of multivariable q-hypergeometric series with respect to parameters, Asian-Eur. J. Math., 7, 2 (2014) · Zbl 1298.33028 · doi:10.1142/S1793557114500193 |
| [26] | Sahai, V.; Verma, A., nth-order q-derivatives of Srivastava’s general triple q-hypergeometric series with respect to parameters, Kyungpook Math. J., 56, 911-925 (2016) · Zbl 1372.33005 · doi:10.5666/KMJ.2016.56.3.911 |
| [27] | Sahai, V.; Verma, A., Recursion formulas for q-hypergeometric and q-Appell series, Commun. Korean Math. Soc., 33, 1, 207-236 (2018) · Zbl 1396.33032 |
| [28] | Shehata, A., On the \((p,q)\)-Humbert functions from the view point of the generating functions method, J. Funct. Spaces, 2020 (2020) · Zbl 1448.33008 |
| [29] | Shehata, A.: On the \((p,q)\)-Bessel functions from the view point of the generating functions method. J. Interdiscip. Math., 1-14 (2020) · Zbl 1448.33008 |
| [30] | Srivastava, B., A note on certain bibasic q-Appell and Lauricella series, Glas. Mat., 30, 1, 29-36 (1995) · Zbl 0836.33011 |
| [31] | Srivastava, H. M.; Jain, V. K., q-series identities and reducibility of basic double hypergeometric functions, Can. J. Math., 38, 1, 215-231 (1986) · Zbl 0566.33002 · doi:10.4153/CJM-1986-010-3 |
| [32] | Srivastava, H. M.; Manocha, H. L., A Treatise on Generating Functions (1984), New York: Halsted, New York · Zbl 0535.33001 |
| [33] | Swarttouw, R. F., The contiguous function relations for basic hypergeometric function, J. Math. Anal. Appl., 149, 1, 151-159 (1990) · Zbl 0703.33012 · doi:10.1016/0022-247X(90)90292-N |
| [34] | Verma, A.; Sahai, V., Some recursion formulas for q-Lauricella series, Afr. Math., 30, 9, 1-44 (2019) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
