On Stirling-type pairs and extended Gegenbauer-Humbert-Fibonacci polynomials. (English) Zbl 0795.05003

Bergum, G. E. (ed.) et al., Applications of Fibonacci numbers. Volume 5: Proceedings of the fifth international conference on Fibonacci numbers and their applications, University of St. Andrews, Scotland, July 20-24, 1992. Dordrecht: Kluwer Academic Publishers. 367-377 (1993).
A characterization of Stirling-type pairs is given. Here, a Stirling-type pair is defined as a pair \((A(n,k)\); \(B(n,k))\) of the coefficients in the expansions \[ {1 \over k!} \bigl( a(t) \bigr)^ k=\sum_{k \geq 0} A(n,k) {t^ n \over n!} \quad \text{ and }\quad {1 \over k!} \bigl( b(t) \bigr)^ k = \sum_{k \geq 0} B(n,k) {t^ n \over n!} \] for a couple of formal power series \[ a(t)=t+ \sum_{k \geq 2} a_ kt^ k \quad \text{and }\quad b(t)=t+\sum_{k\geq 2} b_ kt^ k \] such that \(a(b(t)) = b(a(t))=t\); see [L. C. Hsu, Fibonacci Q. 25, 346-351 (1987; Zbl 0632.10011)]. Note that for \((A(n,k), B(n,k))\) to be a Stirling-type pair it is necessary and sufficient that \[ x_ n=\sum^ n_{k=0} A(n,k) y_ k \Leftrightarrow y_ n=\sum^ n_{k=0} B(n,k)x_ k. \] The author proves (Theorem 1) that holding of the (Schlömilch-type) formula \[ (A/B) (n,k)=\sum^{n-k}_{j=0} (-1)^ j {2n-k \choose n-k-j} {n-1+j \choose n-k+j} (B/A) (n-k+j,j) \] is also necessary and sufficient for \((A,B)\) to be a Stirling-type pair. Various (old and new) examples of Stirling-type pairs are given, including those generated by Stirling-type functions (Theorem 3). For example, the generalized Bernoulli numbers \(B_ n^{(k)}\) defined by the relation \[ t^ k \cdot (e^ t-1)^{-k} =\sum_{n \geq 0} B_ n^{(k)} {t^ n \over n!} \] give rise to a Stirling-type pair \((A(n,k), (B(n,k))\) where \[ A(n,k) \dot={n\choose k}B^{(n)}_{n-2}\quad \text{ and }\quad B(n,k) \dot= {n!(n-1)! \over (2n-k)!(k-1)!} \cdot T(2n-k,n) \] with \(T(n,k)\) being the second component in the Stirling-type pair \((S,T)\) defined by the couple of series \(a(t)=\log (1+t)\) and \(b(t)=e^ t-1\).
Next, the extended Gegenbauer-Humbert-Fibonacci polynomials are defined as the coefficients in the expansion \[ (1-mxt + yt^ m)^{-w} = \sum_{n \geq 0} P_ n (m;x,y,w)t^ n, \] with \(m \in \mathbb{N}\), \(w \in \mathbb{C}\), \(x\) and \(y\) as \(\mathbb{C}\)-variables. Among those are Chebyshev polynomials \(U_ n(x) = P_ n (2;x,1,1)\), Legendre polynomials \(P_ n (2;x,1, {1 \over 2})\), etc.
Moreover, Fibonacci numbers \(F_{n+1}\) are obtained as \(F_{n+1}=P_ n (2;{1 \over 2},-1,1)\). Using the reciprocal representation formula [L. C. Hsu, Port. Math. 48, No. 3, 357-361 (1991; Zbl 0742.05005)], numerous new formulas for those polynomials (and numbers) are obtained.
For the entire collection see [Zbl 0781.00009].


05A15 Exact enumeration problems, generating functions
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)