A new formula for the spherical growth series of an amalgamated free product of two infinite cyclic groups (Q1756004)

Let \(G(p,q)\) be the group with presentation \(\langle x, y \mid x^{p} = y^{q} \rangle\) where \(p\), \(q\) are integers such that \(2 \leq p \leq q\). In the paper under review, an explicit formula for the spherical growth series of the group \(G(p,q)\) with respect to the generating set \(\{x,y,x^{-1},y^{-1}\}\) is obtained. The author remark that a similar formula has been previously obtained by \textit{C. P. Gill} [Int. J. Algebra Comput. 9, No. 1, 1--30 (1999; Zbl 1013.20026)]) but the new formula is deduced with independent methods and can be used to compute explicitly the growth series as rational function.