1,885 research outputs found
Inversions relating Stirling, tanh, Lah numbers and an application to Mathematical Statistics
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
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
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
Extensions of the numbers of the second kind and -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 .
These extensions naturally encompass the well known - 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
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
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
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 -Poisson process
We give an example showing that the product and linearization formulas for
the Wick product versions of the -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
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
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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