1,885 research outputs found

    Inversions relating Stirling, tanh, Lah numbers and an application to Mathematical Statistics

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    Inversion formulas have been found, converting between Stirling, tanh and Lah numbers. Tanh and Lah polynomials, analogous to the Stirling polynomials, have been defined and their basic properties established. New identities for Stirling and tangent numbers and polynomials have been derived from the general inverse relations. In the second part of the paper, it has been shown that if shifted-gamma probability densities and negative binomial distributions are matched by equating their first three semi-invariants (cumulants), then the cumulants of the two distributions are related by a pair of reciprocal linear combinations equivalent to the inversion formulas established in the first part.Comment: 11 page

    On umbral extensions of Stirling numbers and Dobinski-like formulas

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    Umbral extensions of the stirling numbers of the second kind are considered and the resulting dobinski-like various formulas including new ones are presented. These extensions naturally encompass the two well known q-extensions. The further consecutive umbral extensions q-stirling numbers are therefore realized here in a two-fold way. The fact that the umbral q-extended dobinski formula may also be interpreted as the average of powers of random variable with the q-poisson distribution singles out the q-extensions which appear to be a kind of singular point in the domain of umbral extensions as expressed by corresponding two observations. Other relevant possibilities are tackled with the paper`s closing down questions and suggestions with respect to other already existing extensions while a brief limited survey of these other type extensions is being delivered. There the newton interpolation formula and divided differences appear helpful and inevitable along with umbra symbolic language in describing properties of general exponential polynomials of touchard and their possible generalizations. Exponential structures or algebraically equivalent prefabs with their exponential formula appear to be also naturally relevant.Comment: 40 page

    Series with Hermite Polynomials and Applications

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    We obtain a series transformation formula involving the classical Hermite polynomials. We then provide a number of applications using appropriate binomial transformations. Several of the new series involve Hermite polynomials and harmonic numbers, Lucas sequences, exponential and geometric numbers. We also obtain a series involving both Hermite and Laguerre polynomials, and a series with Hermite polynomials and Stirling numbers of the second kind.Comment: arXiv admin note: substantial text overlap with arXiv:1006.250

    New formulas for Stirling-like numbers and Dobinski-like formulas

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    Extensions of the StirlingStirling numbers of the second kind and DobinskiDobinski -like formulas are proposed in a series of exercises for graduates. Some of these new formulas recently discovered by me are to be found in the source paper [1] [1]. These extensions naturally encompass the well known qq- extensions. The indicatory references are to point at a part of the vast domain of the foundations of computer science in arxiv affiliation.Comment: 9 pages, presented at the Gian-Carlo Rota Polish Seminar, http://ii.uwb.edu.pl/akk/sem/sem_rota.ht

    Binomial transform and the backward difference

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    We prove an important property of the binomial transform: it converts multiplication by the discrete variable into a certain difference operator. We also consider the case of dividing by the discrete variable. The properties presented here are used to compute various binomial transform formulas involving harmonic numbers, skew-harmonic numbers, Fibonacci numbers, and Stirling numbers of the second kind. Several new identities are proved and some known results are given new short proofs.Comment: The paper is a slight modification of the journal article in Advances and Applications in Discrete Mathematics, 13 (1) (2014), 43-6

    On Noncentral Tanny-Dowling Polynomials and Generalizations of Some Formulas for Geometric Polynomials

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    In this paper, we establish some formulas for the noncentral Tanny-Dowling polynomials including sums of products and explicit formulas which are shown to be generalizations of known identities. Other important results and consequences are also discussed and presented.Comment: 11 page

    Combinatorial proofs of inverse relations and log-concavity for Bessel numbers

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    Let the Bessel number of the second kind B(n,k) be the number of set partitions of [n] into k blocks of size one or two, and let the Bessel number of the first kind b(n,k) be a certain coefficient in n-th Bessel polynomial. In this paper, we show that Bessel numbers satisfy two properties of Stirling numbers: The two kinds of Bessel numbers are related by inverse formulas, and both Bessel numbers of the first kind and the second kind form log-concave sequences. By constructing sign-reversing involutions, we prove the inverse formulas. We review Krattenthaler's injection for the log-concavity of Bessel numbers of the second kind, and give a new explicit injection for the log-concavity of signless Bessel numbers of the first kind.Comment: 9 pages, 4 figure

    Product formulas on posets, Wick products, and a correction for the qq-Poisson process

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    We give an example showing that the product and linearization formulas for the Wick product versions of the qq-Charlier polynomials in (Anshelevich 2004) are incorrect. Next, we observe that the relation between monomials and several families of Wick polynomials is governed by "incomplete" versions of familiar posets. We compute M\"obius functions for these posets, and prove a general poset product formula. These provide new proofs and new inversion and product formulas for Wick product versions of Hermite, Chebyshev, Charlier, free Charlier, and Laguerre polynomials. By different methods, we prove inversion formulas for the Wick product versions of the free Meixner polynomials.Comment: v4: presentation improvements following comments by the referee. v3: removed the product formula for the free Meixner Wick products, and its linearization corollary. v2: added a result on the isomorphism between incomplete non-crossing matchings and partition

    The number of direct-sum decompositions of a finite vector space

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    The theory of q-analogs develops many combinatorial formulas for finite vector spaces over a finite field with q elements--all in analogy with formulas for finite sets (which are the special case of q=1). A direct-sum decomposition of a finite vector space is the vector space analogue of a set partition. This paper develops the formulas for the number of direct-sum decompositions that are the q-analogs of the formulas for: (1) the number of set partitions with a given number partition signature; (2) the number of set partitions of an n-element set with m blocks (the Stirling numbers of the second kind); and (3) for the total number of set partitions of an n-element set (the Bell numbers)

    Multiple Bracket Function, Stirling Number, and Lah Number Identities

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    The author has constructed multiple analogues of several families of combinatorial numbers in a recent article, including the bracket symbol, and the Stirling numbers of the first and second kind. In the present paper, a multiple analogue of another sequence, the Lah numbers, is developed, and certain associated identities and significant properties of all these sequences are constructed.Comment: 33 page
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