A probabilistic interpretation of a sequence related to Narayana polynomials

A sequence of coefficients appearing in a recurrence for the Narayana polynomials is generalized. The coefficients are given a probabilistic interpretation in terms of beta distributed random variables. The recurrence established by M. Lasalle is then obtained from a classical convolution identity. Some arithmetical properties of the generalized coefficients are also established

Recurrence Relations for Strongly q-Log-Convex Polynomials

We consider a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly q-log-convex. We also prove that the Bessel transformation preserves log-convexity.Comment: 15 page

Classical Scale Mixtures of Boolean Stable Laws

We study Boolean stable laws, $\mathbf{b}_{\alpha,\rho}$, with stability index $\alpha$ and asymmetry parameter $\rho$. We show that the classical scale mixture of $\mathbf{b}_{\alpha,\rho}$ coincides with a free mixture and also a monotone mixture of $\mathbf{b}_{\alpha,\rho}$. For this purpose we define the multiplicative monotone convolution of probability measures, one is supported on the positive real line and the other is arbitrary. We prove that any scale mixture of $\mathbf{b}_{\alpha,\rho}$ is both classically and freely infinitely divisible for $\alpha\leq1/2$ and also for some $\alpha>1/2$. Furthermore, we show the multiplicative infinite divisibility of $\mathbf{b}_{\alpha,1}$ with respect classical, free and monotone convolutions. Scale mixtures of Boolean stable laws include some generalized beta distributions of second kind, which turn out to be both classically and freely infinitely divisible. One of them appears as a limit distribution in multiplicative free laws of large numbers studied by Tucci, Haagerup and M\"oller. We use a representation of $\mathbf{b}_{\alpha,1}$ as the free multiplicative convolution of a free Bessel law and a free stable law to prove a conjecture of Hinz and M{\l}otkowski regarding the existence of the free Bessel laws as probability measures. The proof depends on the fact that $\mathbf{b}_{\alpha,1}$ has free divisibility indicator 0 for $1/2<\alpha$.Comment: 37 page

Recurrence Relations for Strongly q-Log-Convex Polynomials

We consider a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly q-log-convex. We also prove that the Bessel transformation preserves log-convexity.Comment: 15 page

Polynomial Triangles Revisited

A polynomial triangle is an array whose inputs are the coefficients in integral powers of a polynomial. Although polynomial coefficients have appeared in several works, there is no systematic treatise on this topic. In this paper we plan to fill this gap. We describe some aspects of these arrays, which generalize similar properties of the binomial coefficients. Some combinatorial models enumerated by polynomial coefficients, including lattice paths model, spin chain model and scores in a drawing game, are introduced. Several known binomial identities are then extended. In addition, we calculate recursively generating functions of column sequences. Interesting corollaries follow from these recurrence relations such as new formulae for the Fibonacci numbers and Hermite polynomials in terms of trinomial coefficients. Finally, properties of the entropy density function that characterizes polynomial coefficients in the thermodynamical limit are studied in details.Comment: 24 pages with 1 figure eps include