On the matrix versions of q-zeta function, q-digamma function and q-polygamma function

This paper deals with the [Formula: see text]-analogues of generalized zeta matrix function, digamma matrix function and polygamma matrix function. We also discuss their regions of convergence, integral representations and matrix relations obeyed by them. We also give a few identities involving digamma matrix function and [Formula: see text]-hypergeometric matrix series.

An Analogue of the Thue-Morse Sequence

We consider the finite binary words $Z(n)$, $n \in {\Bbb N}$, defined by the following self-similar process: $Z(0):=0$, $Z(1):=01$, and $Z(n+1):=Z(n)\cdot\overline{Z(n-1)}$, where the dot $\cdot$ denotes word concatenation, and $\overline{w}$ the word obtained from $w$ by exchanging the zeros and the ones. Denote by $Z(\infty)=01110100 \dots$ the limiting word of this process, and by $z(n)$ the $n$'th bit of this word. This sequence $z$ is an analogue of the Thue-Morse sequence. We show that a theorem of Bacher and Chapman relating the latter to a "Sierpiński matrix" has a natural analogue involving $z$. The semi-infinite self-similar matrix which plays the role of the Sierpiński matrix here is the zeta matrix of the poset of finite subsets of ${\Bbb N}$ without two consecutive elements, ordered by inclusion. We observe that this zeta matrix is nothing but the exponential of the incidence matrix of the Hasse diagram of this poset. We prove that the corresponding Möbius matrix has a simple expression in terms of the zeta matrix and the sequence $z$.