Some Theorems on the q-Analogue of the Generalized Stirling Numbers

In this paper, we establish more properties for the q-analogue of the unified generalization of Stirling numbers including the vertical and horizontal recurrence relations, and the rational generating function. This generating function plays an important role in deriving one of the explicit formulas in symmetric function form which will be used in giving combinatorial interpretations of the q-analogue in the context of 0-1 tableau. Moreover, using the combinatorics of 0-1 tableaux, we obtain certain generalization of Carlitz identity. 2000 Mathematics Subject Classification: 05A15, 11B65, 11B73.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  Leonard Carlitz,et al.  $q$-Bernoulli numbers and polynomials , 1948 .

[3]  John Riordan,et al.  Introduction to Combinatorial Analysis , 1958 .

[4]  Waleed A. Al-Salam,et al.  q‐Bernoulli numbers and polynomials , 1958 .

[5]  John Riordan,et al.  Introduction to Combinatorial Analysis , 1959 .

[6]  H. Gould.  The $q$-Stirling numbers of first and second kinds , 1961 .

[7]  H. W. Gould,et al.  Operational formulas connected with two generalizations of Hermite polynomials , 1962 .

[8]  Markos V. Koutras,et al.  Non-central stirling numbers and some applications , 1982, Discret. Math..

[9]  Markos V. Koutras,et al.  On the differences of the generalized factorials at an arbitrary point and their combinatorial applications , 1983, Discret. Math..

[10]  Andrei Z. Broder,et al.  The r-Stirling numbers , 1984, Discret. Math..

[11]  C. Charalambides,et al.  Review of the stirling numbers, their generalizations and Statistical Applications , 1988 .

[12]  Pierre Leroux,et al.  Reduced matrices and q-log-concavity properties of q-Stirling numbers , 1990, J. Comb. Theory A.

[13]  C. Chen,et al.  Principles and Techniques in Combinatorics , 1992 .

[14]  Anne de Médicis,et al.  Generalized Stirling Numbers, Convolution Formulae and p, q-Analogues , 1995, Canadian Journal of Mathematics.

[15]  L. C. Hsu,et al.  A Unified Approach to Generalized Stirling Numbers , 1998 .

[16]  Roberto B. Corcino.  Some Theorems on Generalized Stirling Numbers , 2001, Ars Comb..

[17]  R. Corcino,et al.  A q -Analogue of Generalized Stirling Numbers , 2006, The Fibonacci Quarterly.

[18]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.