A Unified Approach to Generalized Stirling Numbers

Hsu, L. C.; Shiue, Peter Jau-Shyong

doi:10.1006/aama.1998.0586

Cited by 134 publications

(126 citation statements)

References 19 publications

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“…For instance, the unified generalization of Stirling numbers S(n, k; α, β, γ) of Hsu and Shuie [7] which is defined by…”

Section: P Q-analogue Of Newton's Interpolation Formulamentioning

confidence: 99%

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ON p, q-DIFFERENCE OPERATOR

Corcino¹,

Montero²

2012

Journal of the Korean Mathematical Society

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Abstract. In this paper, we define a p, q-difference operator and obtain an explicit formula which is used to express the p, q-analogue of the unified generalization of Stirling numbers and its exponential generating function in terms of the p, q-difference operator. Explicit formulas for the non-central q-Stirling numbers of the second kind and non-central q-Lah numbers are derived using the new q-analogue of Newton's interpolation formula. Moreover, a p, q-analogue of Newton's interpolation formula is established.

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“…For instance, the unified generalization of Stirling numbers S(n, k; α, β, γ) of Hsu and Shuie [7] which is defined by…”

Section: P Q-analogue Of Newton's Interpolation Formulamentioning

confidence: 99%

“…The unified generalization of Stirling numbers of Hsu and Shuie [7], denoted by S(n, k; α, β, γ), is expressed explicitly in [4] as (1) S(n, k; α, β, γ)…”

Section: Introductionmentioning

confidence: 99%

ON p, q-DIFFERENCE OPERATOR

Corcino¹,

Montero²

2012

Journal of the Korean Mathematical Society

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“…define polynomials in α, known as generalized Stirling numbers [73,74,20,38] which can be expressed as…”

Section: Combinatorial Identities Related To Random Partitionsmentioning

confidence: 99%

“…In particular, for c = (a + b)/2 the density f a,b := f a,b,(a+b)/2 of β a,b 2Γ (a+b)/2 reduces by the gamma duplication formula (38) to…”

Section: Lemma 13mentioning

confidence: 99%

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Brownian Motion, Bridge, Excursion, and Meander Characterized by Sampling at Independent Uniform Times

Pitman

1999

Electron. J. Probab.

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For a random process X consider the random vector defined by the values of X at times 0 < U n,1 < ... < U n,n < 1 and the minimal values of X on each of the intervals between consecutive pairs of these times, where the U n,i are the order statistics of n independent uniform (0, 1) variables, independent of X. The joint law of this random vector is explicitly described when X is a Brownian motion. Corresponding results for Brownian bridge, excursion, and meander are deduced by appropriate conditioning. These descriptions yield numerous new identities involving the laws of these processes, and simplified proofs of various known results, including Aldous's characterization of the random tree constructed by sampling the excursion at n independent uniform times, Vervaat's transformation of Brownian bridge into Brownian excursion, and Denisov's decomposition of the Brownian motion at the time of its minimum into two independent Brownian meanders. Other consequences of the sampling formulae are Brownian representions of various special functions, including Bessel polynomials, some hypergeometric polynomials, and the Hermite function. Various combinatorial identities involving random partitions and generalized Stirling numbers are also obtained.

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The Group of Generalized Stirling Numbers

Bickel

2001

Advances in Applied Mathematics

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In this paper we provide an algebraic approach to the generalized Stirling numbers (GSN). By defining a group that contains the GSN, we obtain a unified interpretation for important combinatorial functions like the binomials, Stirling numbers, Gaussian polynomials. In particular we show that many GSN are products of others. We provide an explanation for the fact that many GSN appear as pairs and the inverse relations fulfilled by them. By introducing arbitrary boundary conditions, we show a Chu-Vandermonde type convolution formula for GSN.Using the group we demonstrate a solution to the problem of finding the connection constants between two sequences of polynomials with persistent roots.

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A Unified Approach to Generalized Stirling Numbers

Cited by 134 publications

References 19 publications

ON p, q-DIFFERENCE OPERATOR

ON p, q-DIFFERENCE OPERATOR

Brownian Motion, Bridge, Excursion, and Meander Characterized by Sampling at Independent Uniform Times

The Group of Generalized Stirling Numbers

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