zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A unified approach to generalized Stirling numbers. (English) Zbl 0913.05006
Set $(z| \alpha)_n =z(z-\alpha) \cdots (z-n \alpha+ \alpha)$ and $(z | \alpha)_0=1$. For complex parameters $(\alpha, \beta,r) \ne(0,0,0)$, the paper studies the inverse relations $$(t|\alpha)_n=\sum^n_{k=0} S^1(n,k) (t-r | \beta)_k \quad \text{and} \quad (t | \beta)_n =\sum^n_{k=0} S^2(n,k) (t+r | \alpha)_k.$$ The resulting generalized Stirling numbers include binomial coefficients, Lah numbers, signless Stirling numbers, and many other generalized Stirling numbers. Generating functions and Dobinski-type formulae are also given.

MSC:
05A15Exact enumeration problems, generating functions
11B73Bell and Stirling numbers
WorldCat.org
Full Text: DOI
References:
[1] Broder, A. Z.: Ther. Discrete math. 49, 241-259 (1984)
[2] Carlitz, L.: Degenerate Stirling, Bernoulli and Eulerian numbers. Utilitas math. 15, 51-88 (1979) · Zbl 0404.05004
[3] Carlitz, L.: Weighted Stirling numbers of the first and second kind--I, II. Fibonacci quart. 18, 147-162 (1980) · Zbl 0428.05003
[4] Charalambides, C. A.; Koutras, M.: On the differences of the generalized factorials at an arbitrary point and their combinatorial applications. Discrete math. 47, 183-201 (1983) · Zbl 0539.05009
[5] Charalambides, C. A.: On weighted Stirling and other related numbers and some combinatorial applications. Fibonacci quart. 22, 296-309 (1984) · Zbl 0556.05005
[6] Charalambides, C. A.; Singh, J.: A review of the Stirling numbers, their generalizations and statistical applications. Comm. statist. Theory methods 17, 2533-2595 (1988) · Zbl 0696.62025
[7] Comtet, L.: Advanced combinatorics. (1974) · Zbl 0283.05001
[8] Doubilet, P.; Rota, G. -C.; Stanley, R.: On the foundations of combinatorial theory (VI): the idea of generating function. Sixth Berkeley symposium on mathematical statistics and probability 2 (1972) · Zbl 0267.05002
[9] Gould, H. W.; Hopper, A. T.: Operational formulas connected with two generalizations of Hermit polynomials. Duke math. J. 29, 51-63 (1962) · Zbl 0108.06504
[10] Howard, F. T.: A theorem relating potential and Bell polynomials. Discrete math. 39, 129-143 (1982) · Zbl 0478.05008
[11] Howard, F. T.: Degenerate weighted Stirling numbers. Discrete math. 57, 45-58 (1985) · Zbl 0606.10009
[12] Hsu, L. C.: Power-type generating functions. (1990) · Zbl 0768.41030
[13] L. C. Hsu, H. Yu, A unified approach to a class of Stirling-type pairs, Dalian University of Technology, China, 1996
[14] Joni, S. A.; Rota, G. -C.; Sagan, B.: From sets to functions: three elementary examples. Discrete math. 37, 193-202 (1981) · Zbl 0469.05006
[15] Koutras, M.: Non-central Stirling numbers and some applications. Discrete math. 42, 73-89 (1982) · Zbl 0506.10009
[16] Moser, L.; Wyman, M.: Stirling numbers of the second kind. Duke math. J. 25, 29-43 (1958) · Zbl 0079.09102
[17] Moser, L.; Wyman, M.: Asymptotic development of Stirling numbers of the first kind. J. London math. Soc. 33, 133-146 (1958) · Zbl 0081.28202
[18] Mullin, R.; Rota, G. -C.: On the foundations of combinatorial theory. III. theory of binomial enumeration. (1970) · Zbl 0259.12001
[19] Nandi, S. B.; Dutta, S. K.: On associated and generalized lah numbers and applications to discrete distribution. Fibonacci quart., 128-136 (1987) · Zbl 0617.62010
[20] Riordan, J.: Moment recurrence relations for binomial Poisson and hypergeometric frequency distribution. Ann. math. Statist. 8, 103-111 (1937) · Zbl 63.1084.04
[21] Roman, S. M.; Rota, G. -C.: The umbral calculus. Adv. math. 27, 95-188 (1978) · Zbl 0375.05007
[22] Rota, G. -C.: The number of partitions of a set. Amer. math. Monthly 71, 111-117 (1964) · Zbl 0121.01803
[23] Temme, N. M.: Asymptotic estimates of Stirling numbers. Stud. appl. Math. 89, 233-243 (1993) · Zbl 0784.11007
[24] Théorêt, P.: Fonctions, génératrices pour une classe d’equations aux differénces partielles. Ann. sci. Math. Québec 19, 91-105 (1995)
[25] Todorov, P. G.: Taylor expansions of analytic functions related to (1+zx-1. J. math. Anal. appl. 132, 264-280 (1988) · Zbl 0646.30002
[26] Tsylova, E. G.: The asymptotic behavior of generalized Stirling numbers. (1985) · Zbl 0621.60027
[27] Wagner, C. G.: Surjections, differences, and binomial lattices. Stud. appl. Math. 93, 15-27 (1994) · Zbl 0812.05001