Appell polynomial sequences with respect to some differential operators

Abstract

We present a study of a specific kind of lowering operator, herein called Λ, which is defined as a finite sum of lowering operators, proving that this configuration can be altered, for instance, by the use of Stirling numbers. We characterize the polynomial sequences fulfilling an Appell relation with respect to Λ, and considering a concrete cubic decomposition of a simple Appell sequence, we prove that the polynomial component sequences are Λ-Appell, with Λ defined as previously, although by a three term sum. Ultimately, we prove the non-existence of orthogonal polynomial sequences which are also Λ-Appell, when Λ is the lowering operator2 D, where a 0 , a 1 and a 2 are constants and a 2 = 0. The case where a 2 = 0 and a 1 = 0 is also naturally recaptured.

Around operators not increasing the degree of polynomials

“…The technical approach has been already tackled in previous contributions where the authors efforts were mainly focused on an Appel behavior (e.g. [20]). We shall now proceed with the introduction of a wider definition which assembles all the referred situations.…”

“…n = 0, n ≥ 0, matching (2.5), so that J is an isomorphism. If k = 1, then J imitates the usual derivative and is commonly called a lowering operator [6,11,20]. Given a MPS {P n } n≥0 and a non-negative integer k, let us define its (normalized) J-image sequence of polynomials as follows, and notate its dual sequence by {ũ n } n≥0 .…”

“…Remark 4.1. In order to compare the above conclusions with the treatment of the same problem given in [6] and [20] for the operator Λ = σ 0 D +σ 1 DxD, we must first write this Λ in the J format. A few calculations yield…”

“…With this choice, we may ponder the question above for three orders of differentiation of J: k = 0, k = 1 and k = 2. When k = 0, our debate goes around the polynomial coefficients of the differential equation fulfilled by the four family of classic MOPSs, whereas when k = 1, the solution belongs to the classic Laguerre family enlarging the prior results of [6,20], and finally when k = 2, the single solution is indeed an Hermite form. A brief introduction to the analysis of a two-orthogonal MPS is provided in section 5 for a lowering operator of order one.…”

“…In other important contributions like [9], Kwon et al consider a generic differential operator L N [y] = N i=1 a i (x)y (i) (x) and discuss the existence of orthogonal polynomial sequences {P n } n≥0 , with deg (P n (x)) = n, such that L N [P n ](x) = λ n P n , for each n. It is then worth notice that in view of the actual state of art, with regard to differential operators L such that deg (L (P n )) = n, operating into an orthogonal sequence, we have already some acute results, as for instance, the nonexistence of orthogonal solutions among such differential equations of odd order [9]. More recently, some incursions on the study of an Appell-type behavior of orthogonal polynomials [6,20] allowed to gain new insights concerning differential relations fulfilled by orthogonal polynomial sequences.…”