The second‐order self‐associated orthogonal sequences

Abstract

The aim of this work is to describe the orthogonal polynomials sequences which are identical to their second associated sequence. The resulting polynomials are semiclassical of class s ≤ 3. The characteristic elements of the structure relation and of the second-order differential equation are given explicitly. Integral representations of the corresponding forms are also given. A striking particular case is the case of the so-called electrospheric polynomials.

Characterization of basic sets of orthogonalq-polynomials via someq-difference equations

“…Such a sequence is characterized by its quadratic decomposition given by [10,12] W 2n ðxÞ ¼ P n ðx 2 Þ; W 2nþ1 ðxÞ ¼ ðx 2 1ÞR n ðx 2 Þ; n $ 0; ð1:12Þ…”

On the extended connection coefficients between two orthogonal polynomial sequences

“…where {P (r) n } n≥0 is the associated MOPS of order r ≥ 1 (see [1] Let us consider {P n } n≥0 be a second-order self-associated polynomials sequence satisfying (1.1) with [25] ξ n = (−1) n , α n+1 = α, α ∈ C * n ≥ 0.…”

Characterizations of a special family of Hq-semiclassical orthogonal q-polynomials of class one

“…It is possible to associate with the sequence { } ≥0 two MOPSs ≥0 with respect to and { } ≥0 with respect to = 1 1 ( − 1)( ) ful lling the recurrence relation (2.4) with (see [5,6,14,18])…”

“…From 1985, this theory has been developed by P. Maroni from an algebraic aspect and a distributional one [15]. Specially, many authors have highlighted some processes of construction of -semiclassical orthogonal polynomials of class greater than one via the quadratic decomposition [2,6,10,18], the resolution of the nonlinear system (Laguerre-Freud equations) satis ed by the coe cients of the three-term recurrence relation of a such sequence in some particular cases [1,17] and the study of functional equation of the type ( ) = ( ) , where , are two polynomials adequately chosen and , are two forms (linear functional) [10].…”