A \(q\)-analog of Newton’s series, Stirling functions and Eulerian functions. (English) Zbl 0816.33010
In a series of papers, P. L. Butzer, M. Haus and M. Schmidt [Result Math. 16, No. 1/2, 16-48 (1989; Zbl 0707.05002); P. L. Butzer and M. Haus, Stud. Appl. Math. 84, No. 1, 71-91 (1991; Zbl 0738.11025); M. Haus and M. Schmidt, Aequationes Math. 46, No. 1/2, 119-142 (1993; Zbl 0797.11025)] studied extensions to complex arguments of classical combinatorial functions, such as Stirling numbers and Eulerian numbers. In the paper under review, the project of doing the same with the corresponding \(q\)-analogues is started. For \(\alpha\in \mathbb{C}\) and \(n\in \mathbb{N}\) the authors introduce the \(q\)-Stirling function \(S_ q(\alpha, n)\) of the second kind and the \(q\)-Eulerian function \(A_ q(\alpha, n)\), which for \(\alpha\in \mathbb{N}\) agree with the usual \(q\)-Stirling numbers of the second kind and the usual \(q\)-Eulering numbers, respectively. The authors derive several properties and prove various identities for these functions. An example is the identity \(\Gamma_ q(x+ 1)= \sum_{s\geq 0} A_ q(x,s)\), provided \(\text{Re } x>-1\), where \(\Gamma_ q(z)\) is the usual \(q\)-gamma function.
Reviewer: Ch.Krattenthaler (Wien)
MSC:
| 33D05 | \(q\)-gamma functions, \(q\)-beta functions and integrals |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
| 05A19 | Combinatorial identities, bijective combinatorics |
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
| 11B73 | Bell and Stirling numbers |
| 33B15 | Gamma, beta and polygamma functions |
References:
| [1] | Butzer (P.) and Hauss (M.). – On Stirling functions of the second kind, Stud. Appl. Math., t. 84, 1991, p. 71–91. · Zbl 0738.11025 |
| [2] | Butzer (P.) and Hauss (M.). – Eulerian numbers with fractional order parameters, Aequationes Math., t. 1/2, 1993, p. 119–142. · Zbl 0797.11025 · doi:10.1007/BF01834003 |
| [3] | Butzer (P.), Hauss (M.) and Schmidt (M.). – Factorial functions and Stirling numbers of fractional orders, Results in Math., t. 16, 1989, p. 16–48. · Zbl 0707.05002 · doi:10.1007/BF03322642 |
| [4] | Carlitz (L.). – q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc, t. 76, 1954, p. 332–350. · Zbl 0058.01204 |
| [5] | Carlitz (L.). – A note on q-Eulerian numbers, J. Comb. Th. A, t. 25, 1978, p. 90–94. · Zbl 0386.05003 · doi:10.1016/0097-3165(78)90038-9 |
| [6] | Comtet (L.). – Analyse Combinatoire (Vol. 2). – Presses Universitaires de France, Paris, 1970. |
| [7] | Foata (D.) and Zeilberger (D.). – Multibasic Eulerian polynomials, Trans. Amer. Math. Soc, t. 328, 1991, p. 843–862. · Zbl 0790.05003 · doi:10.1090/S0002-9947-1991-1022166-4 |
| [8] | Garsia (A.). –On the ”maj” and ”inv” q-analogues of Eulerian numbers, J. Lin. Multilin. Alg., t- 8, 1979, P. 21–34. · Zbl 0442.05002 · doi:10.1080/03081087908817296 |
| [9] | Gasper (G.) and Rahman (M.). – Basic Hypergeometric Series. – Cambridge Univ. Press, 1990 · Zbl 0695.33001 |
| [10] | Gelfond (A.O.). – Calcul des différences finies. – Dunod, Paris, 1963. |
| [11] | Knopp (K.). – Theory and Application of Infinite Series. – Dover Publications, INC., New York, 1990. · Zbl 0731.11029 |
| [12] | Koornwinder (T. H.). – Jacobi functions as limit cases of q-Ultraspherical polynomials, J. Math. Ana. Appl., t. 148, 1990, p. 44–54. · Zbl 0713.33010 · doi:10.1016/0022-247X(90)90026-C |
| [13] | Milne-Thomson (L. M.). – The calculus of finite differences. – Macmillan and Co Ltd., New York, 1960. · JFM 59.1111.01 |
| [14] | Ramis (J. P.). – About the growth of entire functions solutions of linear algebraic q–difference equations, Annales de la Faculté des Sciences de Toulouse, Série 6, t. 1, 1992, p. 53–94. · Zbl 0796.39005 · doi:10.5802/afst.739 |
| [15] | Rawlings (D.). – Permutation and Multipermutation statistics, Publ. de l’IRMA, Strasbourg, 1979/P-23. |
| [16] | Titchmarsh (E. C). – The Theory of Functions (Second Edition). – Oxford Univ. Press, London - New York, 1948. · Zbl 0005.21004 |
| [17] | Wallisser (R.). –: Uber ganze Funktionen, die in einer geometrischen Folge ganze Werte annehmen, Monatsh. für Math., t. 100, 1985, p. 329–335. · Zbl 0596.30041 · doi:10.1007/BF01339232 |
| [18] | Zeng (J.). – The q-Stirling numbers, continued fractions and the q-Charlier and q-Laguerre polynomials, J. of Comp. and Applied. Math., to appear. · Zbl 0828.30003 |
| [19] | Zhang (C). – La fonction Gamma considérée comme la somme de nombres eulériens généralisés, Europ. J. Combi., to appear. |
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