Let G be a group generated by r elements $g_1,\dots ,g_r$. Among the reduced words in $g_1,\dots ,g_r$ of length n some, say ${\gamma }_n$, represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of ${\gamma }_{2n}$ has a limit, called the cogrowth exponent with respect to the generators $g_1,\dots ,g_r$. We show by analytic methods that the numbers ${\gamma }_n$ vary regularly, i.e. the ratio ${\gamma }_{2n+2}/{\gamma }_{2n}$ is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function associated...

The reciprocal super Catalan matrix has entries [...] . Explicit formulæ for its LU-decomposition, the LU-decomposition of its inverse, and some related matrices are obtained. For all results, q-analogues are also presented.

Bringmann, Lovejoy, and Osburn (2009, 2010) showed that the generating functions of the spt-overpartition functions $\overline{spt}\left(n\right)$, ${\overline{spt}}_1\left(n\right)$, ${\overline{spt}}_2\left(n\right)$, and M2spt(n) are quasimock theta functions, and satisfy a number of simple Ramanujan-like congruences. Andrews, Garvan, and Liang (2012) defined an spt-crank in terms of weighted vector partitions which combinatorially explain simple congruences modulo 5 and 7 for spt(n). Chen, Ji, and Zang (2013) were able to define this spt-crank in terms of ordinary partitions. In this...

Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv 1(mod2m)$ and $2$ is an $m$th power residue modulo $p$, we determine the value of the product ${\prod }_{k\in R_m\left(p\right)}(1+tan(\pi ak/p))$, where $$R_m\left(p\right)=\{0<k<p:k\in \mathbb{Z}\text{is}\text{an}m\text{th}\text{power}\text{residue}\text{modulo}p\}.$$ In particular, if $p=x^2+64y^2$ with $x,y\in \mathbb{Z}$, then $$\prod _{k\in R_4\left(p\right)}\left(1+tan\pi \frac{ak}{p}\right)={(-1)}^y{(-2)}^{(p-1)/8}.$$

Infinite lower triangular matrices of generalized Schröder numbers are used to construct a two-parameter class of invertible sequence transformations. Their inverses are given by triangular matrices of coordination numbers. The two-parameter class of Schröder transformations is merged into a one-parameter class of stretched Riordan arrays, the left-inverses of which consist of matrices of crystal ball numbers. Schröder and inverse Schröder transforms of important sequences are calculated.