Quadratic decomposition of a family ofH_q-semiclassical orthogonal polynomial sequences

Abstract

We deal with monic orthogonal polynomial sequences (MOPSs), {ß,,},,^,,, satisfying the three-term recurrence relation 5,,+2(x) = (x -;S"^.|)ß,,+i(x) -7"+|ß"(x), n = O, 1,2, ..., with initial conditions B^ix) = 1 and S|(jr) = x -ßo, where ß" = (-l)"jßo and y,, ¥" O for all n & 1. These sequences are characterized by the relation ß2"(x) = P"ix^), n > O, ß|(x) = A--A), where {P,,),,ao is a MOPS. In this paper, we show that the sequence {S,,],,20 is W^-semiclassical if and only if the sequence {P,,},,^() is //^2-semiclassical. Then, we express the characteristic elements of the //(^-.semiclassical sequence (fi,, },,2(,, .such as the ç-Pearson equation satisfied by the corresponding linear functional, the class ofthe linear functional, the first-order linear q-difference equation satisfied by the Stieltjes function and the coefficients of the structure relation for such a sequence of polynomials, in terms of the characteristic elements of the sequence, {f,,},,a.(). In particular, if the .sequence {i*,,},,=;() is //^2-semiclassical of class zero, then we obtain a new non-symmetric //,-semiclassical sequence of polynomials \B" },,>o of class .5=1.

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

“…Moreover, the classi cation of the symmetric -semiclassical orthogonal -polynomials of class one was given in [8] by investigating the quadratic operator . For other works in the domain of semiclassical orthogonal polynomials with respect to other operators see [4,16,20].…”

Symbolic Approach to the General Quadratic Polynomial Decomposition

“…If we are dealing with an orthogonal sequence the first fundamental property to consider is the regularity [16,17,21,24]. We can also study the same problem for specific properties like semi-classical [5,29], Laguerre-Hahn [3,4] or Appell characters [14], among others. Also, the decomposition can reveal interesting connections between a sequence and its components and allows a deep study of all families involved as it was the case in [15,32].…”