The authors compute the coefficients \(\lambda_ n\) and \(b_ n\) of the three-term recurrence relation \(P_{n+1}= (x- b_ n) P_ n+ \lambda_ n P_{n-1}\), satisfied by the orthogonal polynomials \(P_ n(x)\) with the \(n\)th moment \(\mu_ n= [1; \beta]_ q [2; \beta]_ q\cdots [n; \beta]_ q\), where \([n;\beta]_ q= \beta+ q+ q^ 2+\cdots+ q^{n-1}\). These polynomials form a family of \(q\)-analogues of the usual Laguerre polynomials \(L^{(\beta- 1)}(x)\).
The authors prove that \(\lambda_ n= q^{2n- 1} [n; 1]_ q [n;\beta]_ q\) and \(b_ n= q^ n([n; 1]_ q+ [n+ 1; \beta]_ q)\) by combinatorial arguments using a former work of the second author [Lect. Notes Math. 1171, 139-157 (1985; Zbl 0588.05006)] and a bijection given by D. Foata and D. Zeilberger [Stud. Appl. Math. 83, No. 1, 31-59 (1990; Zbl 0738.05001)] between the symmetric group on \(n\) elements and some set of combinatorial histories of length \(n\).