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Some notes on the \((q, t)\)-Stirling numbers. (English) Zbl 1403.05020

Summary: The orthogonality of the \((q, t)\)-version of the Stirling numbers has recently been proved by Y. Cai and M. A. Readdy [Adv. Appl. Math. 86, 50–80 (2017; Zbl 1358.05031)] using a bijective argument. In this paper, we introduce new recurrences for the \((q, t)\)-Stirling numbers and provide a \((q, t)\)-analogue for sums of powers. Specializations of these results are given in terms of Stirling numbers or \(q\)-Stirling numbers.

MSC:

05A30 \(q\)-calculus and related topics
11B73 Bell and Stirling numbers

Citations:

Zbl 1358.05031
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References:

[1] Bennett, C.; Dempsey, K.; Sagan, B., Partition lattice \(q\)-analogs related to \(q\)-Stirling numbers, J. Algebraic Combin., 3, 261-283 (1994) · Zbl 0849.05004
[2] Cai, Y.; Readdy, M., \(q\)-Stirling numbers: A new view, Adv. Appl. Math., 86, 50-80 (2017) · Zbl 1358.05031
[3] Carlitz, L., On abelian fields, Trans. Amer. Math. Soc., 35, 122-136 (1933) · JFM 59.0188.02
[4] Ehrenborg, R.; Readdy, M., Juggling and applications to \(q\)-analogues, Discrete Math., 157, 107-125 (1996) · Zbl 0859.05010
[5] Ernst, T., \(q\)-Stirling numbers, an umbral approach, Adv. Dyn. Syst. Appl., 3, 2, 251-282 (2008)
[6] Garrett, K. C.; Hummel, K., A combinatorial proof of the sum of \(q\)-cubes, Electron. J. Combin., 11, #R9 (2004) · Zbl 1050.05012
[7] Gould, H. W., The \(q\)-Stirling numbers of the first and second kinds, Duke Math. J., 28, 281-289 (1961) · Zbl 0201.33601
[8] Guo, V. J.W.; Yang, D.-M., A \(q\)-analogue of some binomial coefficient identities of Y. Sun, Electron. J. Combin., 18, #P78 (2011) · Zbl 1230.05012
[9] Guo, V. J.W.; Zeng, J., A \(q\)-analogue of Faulhaber’s formula for sums of powers, Electron. J. Combin., 11, 2, #R19 (2005) · Zbl 1082.05012
[10] Jordan, C., Calculus of Finite Differences (1947), Chelsea: Chelsea New York
[11] Knuth, D. E., Johann Faulhaber and sums of power, Math. Comp., 61, 277-294 (1993) · Zbl 0797.11026
[12] Macdonald, I. G., Symmetric Functions and Hall Polynomials (1995), Clarendon Press: Clarendon Press Oxford · Zbl 0899.05068
[13] Merca, M., A note on \(q\)-Stirling numbers, (Milanović, G. V.; Rassias, M. Th., Analytic Number Theory, Approximation Theory, and Special Functions (2014), Springer: Springer New-York), 239-244 · Zbl 1362.11033
[14] Merca, M., Generalizations of two identities of Guo and Yang, Quaest. Math., 41, 5, 643-652 (2018) · Zbl 1395.05185
[15] Merca, M., A \(q\)-analogue for sums of powers, Acta Arith., 183, 185-189 (2018) · Zbl 1400.05031
[16] Milne, S., A \(q\)-analog of restricted growth functions, Dobinski’s equality, and Charlier polynomials, Trans. Amer. Math. Soc., 245, 89-118 (1978) · Zbl 0402.05007
[17] Park, S., \(P\)-partitions and \(q\)-Stirling numbers, J. Combin. Theory Ser. A, 68, 33-52 (1994) · Zbl 0809.05006
[18] Schlosser, M., \(q\)-Analogues of the sums of consecutive integers, squares, cubes, quarts and quints, Electron. J. Combin., 11, #R71 (2004) · Zbl 1064.33014
[19] Sun, Y., A simple bijection between binary trees and colored ternary trees, Electron. J. Combin., 17, #N20 (2010) · Zbl 1189.05051
[20] Wachs, M.; White, D., \(p, q\)-Stirling numbers and set partition statistics, J. Combin. Theory Ser. A, 56, 27-46 (1991) · Zbl 0732.05004
[21] Warnaar, S. O., On the \(q\)-analogue of the sum of cubes, Electron. J. Combin., 11, #N13 (2004) · Zbl 1071.05010
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