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My Thoughts on the Structure Relation (STR)

June 4, 2012

If D_q is the Askey Wilson operator, then I have been pondering about sequence of monic polynomials \{ P_n \} satisfying D_q (P_n) = \gamma_n P_{n-1}

To be honest, this is only Case One of a certain structure equation mentioned by Al-Salam and Ismail who can be considered as “giants” in this area of Mathematics called Special Functions.  It can be shown,  by comparison of coefficients, that \gamma_n = [n]_q
where here [n]_q is the so-called q-numbers by Medem and Marcellán  (other two giants in this area of Mathematics) given by

[n]_q = \frac { q^{n/2} - q^{-n/2} } { q^{1/2} - q^{-1/2} }  = \frac{q^n-1}{q-1} q^{(1-n)/2}

so these the q-numbers are related to [n] of Hahn by

[n] = [n]_q q^{(n-1)/2 }

We have that

D_q (T_n) = [n]_q U_{n-1}

where T_n is a degree n Chebyshev polynomial of the first type

and  U_n is a degree n Chebyshev polynomial of the second type

From the book of Ismail (referrence [1] below), on page 135 Theorem 13.1.8, we have


one would think that the above (STR) is enjoyed by a polynomial “closely related” to H_n(x|q), the degree n continuous (Rogers) q-Hermite polynomial.

But after intense computation, it turns out that the MOPS  \{ P_n \} that
satisfy the STR above also satisfy the three-term recurrence relations given by

So which MOPS is that?  Is it “closely related” to the continuous q-Hermite polynomial?  I do not know the answer to both the questions.  I do not even know a simpler form of C_n.  All I know is, for my polynomials to be Rogers q-Hermite, all the B_n‘s must be zeros!

[1]  Mourad E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, (ENCYCLOPEDIA OF MATHEMATICS), Cambridge University Press.

(D_q f) (x) = \frac{\delta_q f (x)}{\delta_q x} where x=\cos \theta and \delta_q g(e^{i \theta}) = g(q^{1/2}e^{i \theta}) -g(q^{-{1/2} e^{i \theta})

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