Related papers: Probabilistic bivariate Bell polynomials
We formulate and prove a general recurrence relation that applies to integrals involving orthogonal polynomials and similar functions. A special case are connection coefficients between two sets of orthonormal polynomials, another example…
Derivative polynomials in two variables are defined by repeated differentiation of the tangent and secant functions. We establish the connections between the coefficients of these derivative polynomials and the numbers of interior and left…
We introduce a multivariate analogue of Bernoulli polynomials and give their fundamental properties: difference and differential relations, symmetry, explicit formula, inversion formula, multiplication theorem, and binomial type formula.…
We define the Bernoulli polynomials with a $q$ parameter in terms of $r$-Whitney numbers of the second kind. Some algebraic properties and combinatorial identities of these polynomials are given. Also, we obtain several relations between…
The Appell-type polynomial family corresponding to the simplest non-commutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal non-commutative probability theories (the…
In the paper, the author presents diagonal recurrence relations for the Stirling numbers of the first kind. As by-products, the author also recovers three explicit formulas for special values of the Bell polynomials of the second kind.
The objective of this paper is, in the main, twofold: Firstly, to develop an algebraic setting for dealing with Bell polynomials and related extensions. Secondly, based on the author's previous work on multivariate Stirling polynomials…
If $A(z)=\sum_{n=0}^\infty a_nz^n$ and $B(z)=\sum_{n=0}^\infty b_nz^n$ are two formal power series, with $a_n,b_n\in \mathbb{R}$, the polynomials $(p_n)_n$ defined by the generating function $$ A(z)B(xz)=\sum_{n=0}^\infty p_n(x)z^n $$ are…
We consider measures supported on the bi-circle and review the recurrence relations satisfied by the orthogonal polynomials associated with these measures constructed using the lexicographical or reverse lexicographical ordering. New…
Recently, B\'{e}nyi and the second author introduced two combinatorial interpretations for symmetrized poly-Bernoulli polynomials. In the present study, we construct bijections between these combinatorial objects. We also define various…
By establishing an interesting connection between ordinary Bell polynomials and rational convolution powers, some composition and inverse relations of Bell polynomials as well as explicit expressions for convolution roots of sequences are…
We derive an expression for the generalized Bernoulli numbers in terms of the Bernoulli numbers involving the (exponential) complete Bell polynomials.
In this paper we revisit the work of E.T. Bell concerning partition polynomials in order to introduce the reciprocal partition polynomials. We give their explicit formulas and apply the result to compute closed formulae for some well-known…
We construct the multivariate probability distributions (P\'olya, inverse P\'olya, hypergeometric and negative hypergeometric) from the generalized quantum algebra. Moreover, we derive the bivariate probability distributions and determine…
The series expansion of a power of the modified Bessel function of the first kind is studied. This expansion involves a family of polynomials introduced by C. Bender et al. New results on these polynomials established here include…
We question the commonly accepted statement that random numbers certified by Bell's theorem carry some special sort of randomness, so to say, quantum randomness or intrinsic randomness. We show that such numbers can be easily generated by…
In this paper, we derive some identities involving special numbers and moments of random variables by using the generating functions of the moments of certain random variables. Here the related special numbers are Stirling numbers of the…
Quantum correlations arising in Bell experiments, involving a physical source that emits a quantum state to a number of observers, have been intensively studied over the last decades. Much less is known about the nature of quantum…
We investigate the second-order correlation function of the characteristic polynomial of a sample covariance matrix. Starting from an explicit formula for the generating function, we re-obtain several well-known kernels from random matrix…
Spivey's combinatorial method revealed an important identity for Bell numbers, involving Stirling numbers of the second kind. This paper extends his work by deriving Spivey-type recurrence relations for fully degenerate Bell polynomials and…