Discrete Mathematics, ISSN 0012-365X, 03/2019, Volume 342, Issue 3, pp. 628 - 634

The orthogonality of the (q,t)-version of the Stirling numbers has recently been proved by Cai and Readdy using a bijective argument. In this paper, we...

Stirling numbers | Recurrence relations | [formula omitted]-analogues | [Formula presented]-analogues | MATHEMATICS | Q-STIRLING NUMBERS | Q-ANALOGS | (q, t)-analogues | CUBES | SUMS

Stirling numbers | Recurrence relations | [formula omitted]-analogues | [Formula presented]-analogues | MATHEMATICS | Q-STIRLING NUMBERS | Q-ANALOGS | (q, t)-analogues | CUBES | SUMS

Journal Article

Electronic Notes in Discrete Mathematics, ISSN 1571-0653, 10/2016, Volume 54, pp. 361 - 366

Stirling numbers of the second kind and Bell numbers for graphs were defined by Duncan and Peele in 2009. In a previous paper, one of us, jointly with Nyul,...

q-Stirling numbers | q-Bell numbers | special numbers for graphs | q-Fibonacci numbers | Bell numbers | q-analogues | Stirling numbers

q-Stirling numbers | q-Bell numbers | special numbers for graphs | q-Fibonacci numbers | Bell numbers | q-analogues | Stirling numbers

Journal Article

Applicable Analysis and Discrete Mathematics, ISSN 1452-8630, 04/2018, Volume 12, Issue 1, pp. 178 - 191

We define the (q, alpha)-Whitney numbers which are reduced to the a -Whitney numbers when q -> 1. Moreover, we obtain several properties of these numbers such...

(q, α)-Whitney numbers | α-Whitney-Lah numbers | α-Whitney numbers | Q-Stirling numbers | R-Whitney numbers | alpha -Whitney numbers | MATHEMATICS | q-Stirling numbers | MATHEMATICS, APPLIED | alpha-Whitney-Lah numbers | (q, alpha)-Whitney numbers | BERNOULLI POLYNOMIALS | r-Whitney numbers

(q, α)-Whitney numbers | α-Whitney-Lah numbers | α-Whitney numbers | Q-Stirling numbers | R-Whitney numbers | alpha -Whitney numbers | MATHEMATICS | q-Stirling numbers | MATHEMATICS, APPLIED | alpha-Whitney-Lah numbers | (q, alpha)-Whitney numbers | BERNOULLI POLYNOMIALS | r-Whitney numbers

Journal Article

ELECTRONIC JOURNAL OF COMBINATORICS, ISSN 1077-8926, 04/2019, Volume 26, Issue 2

We exhibit a connection between two statistics on set partitions, the intertwining number and the depth-index. In particular, results link the intertwining...

MATHEMATICS | Q-STIRLING NUMBERS | INVOLUTIONS | CHARLIER | MATHEMATICS, APPLIED | ANALOG | CONGRUENCE B-ORBITS

MATHEMATICS | Q-STIRLING NUMBERS | INVOLUTIONS | CHARLIER | MATHEMATICS, APPLIED | ANALOG | CONGRUENCE B-ORBITS

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 04/2014, Volume 232, pp. 132 - 143

In this paper we derive q-analogues of the multiparameter non-central Stirling numbers of the first and second kind, introduced by El-Desouky. Moreover,...

[formula omitted]-Analogue | Harmonic numbers | Comtet numbers | Stirling numbers | Multiparameter non-central Stirling numbers | [formula omitted]-Stirling numbers | Generalized [formula omitted]-harmonic numbers | q-Stirling numbers | Generalized q-harmonic numbers | q-Analogue | MATHEMATICS, APPLIED | Statistics | Algorithms | Electrical engineering | Harmonics | Matrix representation | Computation | Mathematical analysis | Mathematical models | Joints | Combinatorial analysis

[formula omitted]-Analogue | Harmonic numbers | Comtet numbers | Stirling numbers | Multiparameter non-central Stirling numbers | [formula omitted]-Stirling numbers | Generalized [formula omitted]-harmonic numbers | q-Stirling numbers | Generalized q-harmonic numbers | q-Analogue | MATHEMATICS, APPLIED | Statistics | Algorithms | Electrical engineering | Harmonics | Matrix representation | Computation | Mathematical analysis | Mathematical models | Joints | Combinatorial analysis

Journal Article

Journal of Difference Equations and Applications, ISSN 1023-6198, 12/2008, Volume 14, Issue 12, pp. 1267 - 1277

The main purpose of this paper is to investigate several further interesting properties of symmetry for the p-adic invariant integrals on Zp. From these...

q-Volkenborn integrals | q-Euler numbers | q-stirling numbers | q-Bernoulli numbers

q-Volkenborn integrals | q-Euler numbers | q-stirling numbers | q-Bernoulli numbers

Journal Article

Mathematische Nachrichten, ISSN 0025-584X, 04/2016, Volume 289, Issue 5-6, pp. 693 - 717

We introduce, characterise and provide a combinatorial interpretation for the so‐called q‐Jacobi–Stirling numbers. This study is motivated by their key role in...

q‐Stirling numbers | 05A10 (34B24 34L05) | signed partitions | 33C45 | Orthogonal polynomials | q‐Jacobi–Stirling numbers | q‐classical polynomials | q‐differential equations | q-Stirling numbers | q-classical polynomials | q-differential equations | Signed partitions | q-Jacobi-Stirling numbers | MATHEMATICS | BOCHNER CONDITION | Statistics | Differential equations | Combinatorics | Mathematics

q‐Stirling numbers | 05A10 (34B24 34L05) | signed partitions | 33C45 | Orthogonal polynomials | q‐Jacobi–Stirling numbers | q‐classical polynomials | q‐differential equations | q-Stirling numbers | q-classical polynomials | q-differential equations | Signed partitions | q-Jacobi-Stirling numbers | MATHEMATICS | BOCHNER CONDITION | Statistics | Differential equations | Combinatorics | Mathematics

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 01/2013, Volume 397, Issue 2, pp. 522 - 528

In this paper, by expressing the sums of products of the extended q-Euler polynomials in terms of the special values of the alternating multiple Hurwitz q-zeta...

[formula omitted]-Stirling numbers of the first kind | Alternating multiple Hurwitz [formula omitted]-zeta functions | Alternating Hurwitz [formula omitted]-zeta functions | Extended [formula omitted]-Euler numbers and polynomials | Sums of products | Higher-order extended [formula omitted]-Euler numbers and polynomials | Alternating Hurwitz q-zeta functions | Q-Stirling numbers of the first kind | Extended q-Euler numbers and polynomials | Higher-order extended q-Euler numbers and polynomials | Alternating multiple Hurwitz q-zeta functions | MATHEMATICS | MATHEMATICS, APPLIED | Q-BERNOULLI NUMBERS | q-Stirling numbers of the first kind

[formula omitted]-Stirling numbers of the first kind | Alternating multiple Hurwitz [formula omitted]-zeta functions | Alternating Hurwitz [formula omitted]-zeta functions | Extended [formula omitted]-Euler numbers and polynomials | Sums of products | Higher-order extended [formula omitted]-Euler numbers and polynomials | Alternating Hurwitz q-zeta functions | Q-Stirling numbers of the first kind | Extended q-Euler numbers and polynomials | Higher-order extended q-Euler numbers and polynomials | Alternating multiple Hurwitz q-zeta functions | MATHEMATICS | MATHEMATICS, APPLIED | Q-BERNOULLI NUMBERS | q-Stirling numbers of the first kind

Journal Article

Taiwanese Journal of Mathematics, ISSN 1027-5487, 2/2011, Volume 15, Issue 1, pp. 241 - 257

The main object of this paper is to give -extensions of several explicit relationships of H. M. Srivastava and Á. Pintér [ (2004), 375-380] between the...

Polynomials | Mathematics | Q-bernoulli polynomials | Euler polynomials | Bernoulli polynomials | Recurrence relationships | NUMBERS | q-Extensions | q-Euler polynomials | Kronecker symbol | MATHEMATICS | ZETA | Difference equations | q-Bernoulli numbers | q-Stirling numbers of the second kind | APOSTOL-BERNOULLI | q-Bernoulli polynomials | q-Euler numbers | Addition theorems | FORMULAS

Polynomials | Mathematics | Q-bernoulli polynomials | Euler polynomials | Bernoulli polynomials | Recurrence relationships | NUMBERS | q-Extensions | q-Euler polynomials | Kronecker symbol | MATHEMATICS | ZETA | Difference equations | q-Bernoulli numbers | q-Stirling numbers of the second kind | APOSTOL-BERNOULLI | q-Bernoulli polynomials | q-Euler numbers | Addition theorems | FORMULAS

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 07/2014, Volume 415, Issue 1, pp. 497 - 498

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 10/2009, Volume 215, Issue 3, pp. 1185 - 1208

In this paper, we systematically recover the identities for the q-eta numbers η and the q-eta polynomials η (x), presented by Carlitz [L. Carlitz, q-Bernoulli...

Euler-Maclaurin summation formula | q-Extensions of the Riemann zeta function and the Hurwitz zeta function | q-Stirling numbers of the second kind | q-Extensions of the Bernoulli and Euler polynomials and numbers | MATHEMATICS, APPLIED | Q-ANALOGS | DIRICHLET SERIES

Euler-Maclaurin summation formula | q-Extensions of the Riemann zeta function and the Hurwitz zeta function | q-Stirling numbers of the second kind | q-Extensions of the Bernoulli and Euler polynomials and numbers | MATHEMATICS, APPLIED | Q-ANALOGS | DIRICHLET SERIES

Journal Article

Discrete Mathematics, ISSN 0012-365X, 08/2012, Volume 312, Issue 15, pp. 2197 - 2202

We obtain explicit formulas that express the complete homogeneous symmetric polynomials of the sequence of partial sums sk of a sequence xk as polynomials in...

Legendre–Stirling numbers | Symmetric polynomials | Gaussian coefficients | Generalized Stirling numbers | [formula omitted]-Stirling numbers | q-Stirling numbers | Legendre-Stirling numbers | MATHEMATICS | DIVIDED DIFFERENCES | PASCAL MATRICES

Legendre–Stirling numbers | Symmetric polynomials | Gaussian coefficients | Generalized Stirling numbers | [formula omitted]-Stirling numbers | q-Stirling numbers | Legendre-Stirling numbers | MATHEMATICS | DIVIDED DIFFERENCES | PASCAL MATRICES

Journal Article

Journal of Integer Sequences, 09/2008, Volume 11, Issue 3

Journal Article

Discrete Mathematics, ISSN 0012-365X, 07/2015, Volume 338, Issue 7, pp. 1067 - 1074

We derive a combinatorial equilibrium for bounded juggling patterns with a random, q-geometric throw distribution. The dynamics are analyzed via rook...

Markov process | Combinatorial stationary distribution | Ferrers board | [formula omitted]-Stirling number | Juggling pattern | q-Stirling number | POLYNOMIALS | MATHEMATICS | Q-STIRLING NUMBERS | SEQUENCES

Markov process | Combinatorial stationary distribution | Ferrers board | [formula omitted]-Stirling number | Juggling pattern | q-Stirling number | POLYNOMIALS | MATHEMATICS | Q-STIRLING NUMBERS | SEQUENCES

Journal Article

Axioms, ISSN 2075-1680, 03/2013, Volume 2, Issue 1, pp. 10 - 19

In this paper, we define the generating functions for the generalized q-Stirling numbers of the second kind. By applying Mellin transform to these functions,...

q-Bernoulli numbers and polynomials | Generalized q-Stirling numbers of the second kind | Q-zeta function | generalized q-Stirling numbers of the second kind | q-zeta function

q-Bernoulli numbers and polynomials | Generalized q-Stirling numbers of the second kind | Q-zeta function | generalized q-Stirling numbers of the second kind | q-zeta function

Journal Article

Journal of the Korean Mathematical Society, ISSN 0304-9914, 2010, Volume 47, Issue 3, pp. 645 - 657

In this paper, more generalized q-factorial coefficients are examined by a natural extension of the q-factorial on a sequence of any numbers Tins immediately...

Q-stirling numbers | Q-factorial | Q-lah numbers | MATHEMATICS | q-Stirling numbers | MATRIX | MATHEMATICS, APPLIED | q-factorial | q-Lah numbers

Q-stirling numbers | Q-factorial | Q-lah numbers | MATHEMATICS | q-Stirling numbers | MATRIX | MATHEMATICS, APPLIED | q-factorial | q-Lah numbers

Journal Article

Discrete Mathematics, ISSN 0012-365X, 2010, Volume 310, Issue 17, pp. 2280 - 2298

Multiple q t -binomial coefficients and multiple analogues of several celebrated families of related special numbers are constructed in this paper. These...

Multiple special numbers | Multiple binomial coefficients | Discrete probability measures | Well-poised Jackson coefficients | Well-poised Macdonald functions | Q-STIRLING NUMBERS | CHARLIER | IDENTITIES | PROOF | MATHEMATICS | RESTRICTED GROWTH FUNCTIONS | Q-BERNOULLI | TRANSFORMATIONS | Heterocyclic compounds | Statistics

Multiple special numbers | Multiple binomial coefficients | Discrete probability measures | Well-poised Jackson coefficients | Well-poised Macdonald functions | Q-STIRLING NUMBERS | CHARLIER | IDENTITIES | PROOF | MATHEMATICS | RESTRICTED GROWTH FUNCTIONS | Q-BERNOULLI | TRANSFORMATIONS | Heterocyclic compounds | Statistics

Journal Article

Advances in Applied Mathematics, ISSN 0196-8858, 2007, Volume 38, Issue 3, pp. 275 - 301

Adin, Brenti, and Roichman [R.M. Adin, F. Brenti, Y. Roichman, Descent numbers and major indices for the hyperoctahedral group, Adv. in Appl. Math. 27 (2001)...

Carlitz's identity | Major index | Coinvariant algebra | Descent number | Hilbert series | Hyperoctahedral group | SIGNED PERMUTATION STATISTICS | Q-STIRLING NUMBERS | MATHEMATICS, APPLIED | major index | descent number | CALCULUS | coinvariant algebra | hyperoctahedral group | EULERIAN POLYNOMIALS

Carlitz's identity | Major index | Coinvariant algebra | Descent number | Hilbert series | Hyperoctahedral group | SIGNED PERMUTATION STATISTICS | Q-STIRLING NUMBERS | MATHEMATICS, APPLIED | major index | descent number | CALCULUS | coinvariant algebra | hyperoctahedral group | EULERIAN POLYNOMIALS

Journal Article

Bulletin of the Malaysian Mathematical Sciences Society, ISSN 0126-6705, 2011, Volume 34, Issue 3, pp. 487 - 501

In this paper, we establish more properties for the q-analogue of the unified generalization of Stirling numbers including the vertical and horizontal...

Q-stirling numbers | A-tableau | Exponential-type stirling numbers | Generalized factorial | Rational generating function | Stirling numbers | 0-1 tableau | MATHEMATICS | exponential-type Stirling numbers | q-Stirling numbers | generalized factorial | rational generating function | COMBINATORIAL APPLICATIONS | FORMULAS

Q-stirling numbers | A-tableau | Exponential-type stirling numbers | Generalized factorial | Rational generating function | Stirling numbers | 0-1 tableau | MATHEMATICS | exponential-type Stirling numbers | q-Stirling numbers | generalized factorial | rational generating function | COMBINATORIAL APPLICATIONS | FORMULAS

Journal Article

Electronic Journal of Combinatorics, ISSN 1077-8926, 02/2018, Volume 25, Issue 1

We give combinatorial proofs of q-Stirling identities using restricted growth words. This includes a poset theoretic proof of Carlitz's identity, a new proof...

Poset decomposition | Q-Stirling numbers | Restricted growth words | Q-analogues | q-Stirling numbers | MATHEMATICS, APPLIED | CHARLIER POLYNOMIALS | NUMBERS | Q-ANALOGS | q-analogues | restricted growth words | FROBENIUS | MATHEMATICS | EULER-MAHONIAN STATISTICS | SET PARTITION STATISTICS | RESTRICTED GROWTH FUNCTIONS | poset decomposition | FORMULAS

Poset decomposition | Q-Stirling numbers | Restricted growth words | Q-analogues | q-Stirling numbers | MATHEMATICS, APPLIED | CHARLIER POLYNOMIALS | NUMBERS | Q-ANALOGS | q-analogues | restricted growth words | FROBENIUS | MATHEMATICS | EULER-MAHONIAN STATISTICS | SET PARTITION STATISTICS | RESTRICTED GROWTH FUNCTIONS | poset decomposition | FORMULAS

Journal Article

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