Inversions Relating Stirling, Tanh, Lah Numbers and an Application to Mathematical Statistics

نویسنده

  • GIACOMO DELLA RICCIA
چکیده

Abstract. 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.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

A general two-term recurrence

The recurrence is significant since if we set ai = 0, bi = i for all i we get the Stirling numbers of the second kind, and if we set ai = bi = i for all i we get the Lah numbers. The Lah numbers count the partitions of an n-set into blocks of size k in which the numbers in each block are ordered. The proof in [1] involves generating functions and is rather long. Here we give a short induction p...

متن کامل

Stirling Numbers and Generalized Zagreb Indices

We show how generalized Zagreb indices $M_1^k(G)$ can be computed by using a simple graph polynomial and Stirling numbers of the second kind. In that way we explain and clarify the meaning of a triangle of numbers used to establish the same result in an earlier reference.

متن کامل

Six Proofs for an Identity of the Lah Numbers

In the paper, utilizing respectively the induction, a generating function of the Lah numbers, the Chu-Vandermonde summation formula, an inversion formula, the Gauss hypergeometric series, and two generating functions of Stirling numbers of the first kind, the authors collect and provide six proofs for an identity of the Lah numbers.

متن کامل

On xD-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra W = 〈x,D|Dx − xD = 1〉. Any word ω ∈ W with m x’s and n D’s can be expressed in the normally ordered form ω = xm−n ∑ k>0 { ω k } xkDk, where { ω k } is known as the Stirling number of the second kind for the word ω. This study consid...

متن کامل

A generalized recurrence formula for Stirling numbers and related sequences

In this note, we provide a combinatorial proof of a generalized recurrence formula satisfied by the Stirling numbers of the second kind. We obtain two extensions of this formula, one in terms of r-Whitney numbers and another in terms of q-Stirling numbers of Carlitz. Modifying our proof yields analogous formulas satisfied by the r-Stirling numbers of the first kind and by the r-Lah numbers.

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2004