On Euler’s first transformation formula for \(k\)-hypergeometric function. (English) Zbl 1513.33018
Summary: S. Mubeen and G. M. Habibullah [Int. J. Contemp. Math. Sci. 7, No. 1–4, 89–94 (2012; Zbl 1248.33005); Int. Math. Forum 7, No. 1–4, 203–207 (2012; Zbl 1251.33004)] obtained Kummer’s first transformation for the \(k\)-hypergeometric function. The aim of this note is to provide the Euler-type first transformation for the \(k\)-hypergeometric function. As a limiting case, we recover the results of [loc. cit.]. In addition to this, an alternate and easy derivation of Kummer’s first transformation for the \(k\)-hypergeometric function is also given.
MSC:
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
| 33C15 | Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) |
Keywords:
Euler-type transformation; Kummer-type transformation; hypergeometric function; \(k\)-hypergeometric functionReferences:
| [1] | R. DIAZ and C. TERUEL,q, k-generalized gamma and beta function, Journal of Nonlinear Mathematical Physics, 12, (2005), pp. 118-134. · Zbl 1075.33010 |
| [2] | R. DIAZ and E. PARIGUAN, On hypergeometric functions and Pochhammerk-symbol, Divulgaciones Matematicas, 15, (2007), pp. 179-192. · Zbl 1163.33300 |
| [3] | R. DIAZ, C. ORTIZ and E. PARIGUAN, On thek-gammaq-distribution, Central European Journal of Mathematics, 8, (2010), pp. 448-458. · Zbl 1202.05012 |
| [4] | C. G. KOKOLOGIANNAKI, Properties and inequalities of generalizedk-gamma, beta and zeta functions, Int. J. Contemp. Math. Sciences, 5, (2010), pp. 653-660. · Zbl 1202.33003 |
| [5] | V. KRASNIQI, A limit for thek-gamma andk-beta function, Int. Math. Forum, 5, (2010), pp. 1613-1617. · Zbl 1206.33005 |
| [6] | V. KRASNIQI, Inequalities and monotonicity for the ration ofk-gamma function, Scientia Magna, 6(1), (2010), pp. 40-45. |
| [7] | M. MANSOUR, Determining thek-generalized gamma functionΓk(x)by functional equations, Int. J. Contemp. Math. Sciences, 4, (2009), pp. 1037-1042. · Zbl 1186.33002 |
| [8] | F. MEROVCI, Power product inequalities for theΓkfunction, Int. J. Math. Analysis, 4, (2010), pp. 1007-1012. · Zbl 1207.33001 |
| [9] | S. MUBEEN,k-analogue of Kummer’s first formula, J. Inequal. Spec. Funct., 3(3), (2012), pp. 41-44. · Zbl 1315.33011 |
| [10] | S. MUBEEN and G. M. HABIBULLAH,k-fractional integrals and applications, Int. J. Contemp. Math. Sciences, 7, (2012), pp. 89-94. · Zbl 1248.33005 |
| [11] | S. MUBEEN and G. M. HABIBULLAH,An integral representation of somek-hypergeometric functions, Int. Math. Forum, 7, (2012), pp. 203-207. · Zbl 1251.33004 |
| [12] | E. D. RAINVILLE, Special Functions, The Macmillan Company, New York, (1965). AJMAA, Vol. 18 (2021), No. 2, Art. 3, 6 pp |
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