Generalized Bernoulli polynomials revisited and some other Appell sequences.

*(English)*Zbl 1097.33010In 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. NĂ¶rlund 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.

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.

Reviewer: Stamatis Koumandos (Nicosia)

##### MSC:

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 |