By using a generating function approach it is shown that the sum-of-digits function (related to speciic nite and innnite linear recurrences) satisses a central limit theorem. Additionally a local limit theorem is derived.

Abstract. Let ψ1, . . . , ψk be maps from Z to an additive abelian group with positive periods n1, . . . , nk respectively. We show that the function ψ1 + · · · + ψk is constant if ψ1(x) + · · · + ψk(x) equals a constant for |S| consecutive integers x where S = {r/ns : r = 0, . . . , ns −1; s = 1, . . . , k}. This local-global theorem extends a previous result [Math. Res. Lett. 11(2004), 187–19...

We study basis partitions, introduced by Hansraj Gupta in 1978. For this family of partitions, we give a recurrence, a generating function, identities relating basis partitions to more familiar families of partitions, and a new characterization of basis partitions.

We provide two combinatorial proofs that linear recurrences with constant coefficients have a closed form based on the roots of its characteristic equation. The proofs employ sign-reversing involutions on weighted tilings.

A partition π of the set [n] = {1, 2, . . . , n} is a collection {B1, . . . , Bk} of nonempty pairwise disjoint subsets of [n] (called blocks) whose union equals [n]. In this paper, we find exact formulas and/or generating functions for the number of partitions of [n] with k blocks, where k is fixed, which avoid 3-letter patterns of type x − yz or xy − z, providing generalizations in several in...

The τ-function theory of Painlevé systems is used to derive recurrences in the rank n of certain random matrix averages over U (n). These recurrences involve auxilary quantities which satisfy discrete Painlevé equations. The random matrix averages include cases which can be interpreted as eigenvalue distributions at the hard edge and in the bulk of matrix ensembles with unitary symmetry. The re...

Abstract: Consider lattice paths in the plane allowing the steps (1,1), (1,-1), and (w,0), for some nonnegative integer w. For n > 1, let E(n,0) denote the set of paths from (0,0) to (n,0) running strictly above the x-axis except initially and finally. Generating functions are given for sums of moments of the ordinates of the lattice points on the paths in E(n,0). In particular, recurrencess ar...

For any homomorphism V on the space of symmetric functions, we introduce an operation which creates a q-analog of V. By giving several examples we demonstrate that this quantization occurs naturally within the theory of symmetric functions. In particular, we show that the Hall-Littlewood symmetric functions are formed by taking this q-analog of the Schur symmetric functions and the Macdonald sy...

We define an overpartition analogue of Gaussian polynomials (also known as q-binomial coefficients) as a generating function for the number of overpartitions fitting inside the M ×N rectangle. We call these new polynomials over Gaussian polynomials or over q-binomial coefficients. We investigate basic properties and applications of over q-binomial coefficients. In particular, via the recurrence...

A recurrent formula is presented, for the enumeration of the compositions of positive integers as sums over multisets of positive integers, that closely resembles Euler’s recurrence based on the pentagonal numbers, but where the coefficients result from a discrete integration of Euler’s coefficients. Both a bijective proof and one based on generating functions show the equivalence of the subjec...