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Generalized Bernoulli polynomials revisited and some other Appell sequences. (English) Zbl 1097.33010
In this paper the authors study Appell sequences, that is, sequences of polynomials \(C_{n}(x)\), \(n=0,1,2,\ldots\) satisfying \(C_{n+1}^{\prime}(x)=(n+1)\,C_{n}(x)\), in terms of dual sequences. As special cases of \(C_{n}(x)\) the authors consider the sequences of generalized Bernoulli and Euler polynomials. In particular, the authors consider the following problem: Let \(C_{n}(x)\), \(n=0,1,2,\ldots\) be an Appell sequence, determine all sequences \(B_{n}(x)\), \(n=0,1,2,\ldots\) such that \(B_{n+1}^{\prime}(x)=(n+1)\,B_{n}(x)\) and \(B_{n+1}(x+1)-B_{n+1}(x)=(n+1)\,C_{n}(x)\). The case where \(C_{n}(x)\) is the sequence of generalized Bernoulli polynomials \(B_{n}(x;k)\), has been studied by N. E. Nörlund in the paper [Acta Math. 43, 121–196 (1920; JFM 47.0216.05)].
The authors give new results concerning the dual sequence of the generalized Bernoulli sequence. The dual sequence \(u_{n}\) of a sequence \(P_{n}(x)\) of monic polynomials, is a sequence of elements of the dual space \(\mathcal{P}^{\prime}\) of the vector space \(\mathcal{P}\) of polynomials with coefficients in \(\mathbb{C}\), satisfying \(\langle u_{n}, P_{m}\rangle= \delta_{m,m}\).
In the paper under review, it is shown that the canonical form \(u_{0}(k)\) of a generalized Bernoulli sequence \(B_{n}(x;k)\), \(k\geq 1\), satisfies a homogeneous \(k\)th-order linear equation with polynomial coefficients and is a positive definite form. Similar results are obtained in the case where \(C_{n}(x)\) is the sequence of monic Hermite polynomials \(\hat{H}_{n}(x)\). The case where \(C_{n}(x)\) is the sequence of generalized Euler polynomials \(E_{n}(x;k)\), is also studied and its canonical form \(e_{0}(k)\), \(k\geq 0\) is determined.

33C65 Appell, Horn and Lauricella functions
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
11B68 Bernoulli and Euler numbers and polynomials
JFM 47.0216.05