Some Orthogonal q‐Polynomials

We survey in this paper characterization theorems dealing with polynomial sets which are orthogonal on the real line.

https://link.springer.com/content/pdf/10.1007%2F978-94-009-0501-6_1.pdf

Characterization Theorems for Orthogonal Polynomials

De Moivre gave a simple closed form expression for the mean absolute deviation of the binomial distribution. Later authors showed that similar closed form expressions hold for many of the other classical families. We review the history of these identities and extend them to obtain summation formulas for the expectations of all polynomials orthogonal to the constants.

/pdf/closed-form-summation-for-classical-distributions-variations-20j1yk5g1e.pdf

Closed Form Summation for Classical Distributions: Variations on a Theme of De Moivre

The Muntz-Legendre polynomials arise by orthogonalizing the Muntz system {xIo, x'l~, . .. } with respect to Lebesgue measure on [0, 1] . In this paper, differential and integral recurrence formulae for the Muntz-Legendre polynomials are obtained. Interlacing and lexicographical properties of their zeros are studied, and the smallest and largest zeros are universally estimated via the zeros of Laguerre polynomials. The uniform convergence of the Christoffel functions is proved equivalent to the nondenseness of the Muntz space on [0, 1], which implies that in this case the orthogonal Muntz-Legendre polynomials tend to 0 uniformly on closed subintervals of [0, 1) . Some inequalities for Muntz polynomials are also investigated, most notably, a sharp L2 Markov inequality is proved.

/pdf/muntz-systems-and-orthogonal-muntz-legendre-polynomials-4rdnlzua0x.pdf

Müntz systems and orthogonal Müntz-Legendre polynomials

Recently many people have studied the Sobolev orthogonal polynomials, that is, polynomials which are orthogonal relative to a symmetric bilinear form $\\phi(\\cdot,\\cdot)$ defined by $$ (1.1) $\\phi(p,q) := (p,q)_N = \\sum_{k=0}^{N} \\int_{R}p^(k) (x)q^(k) (x) d\\mu_k, $$ where each $d\\mu_k$ is a signed Borel measure on the real line $R$ with finite moments of all orders. For the brief history on this subject, we refer to the survey article Ronveaux [13] and Marcellan and et al [10].

/pdf/sobolev-orthogonal-polynomials-and-second-order-differential-1epgak2a7u.pdf

Sobolev orthogonal polynomials and second-order differential equations

/pdf/on-a-certain-class-of-orthogonal-polynomials-3l50vuow0c.pdf

On a Certain Class of Orthogonal Polynomials

/pdf/on-the-polynomials-related-to-the-differential-equation-frac-qcdfa5nvns.pdf

Frank S. Beale

Papers

On a Certain Class of Orthogonal Polynomials

On the Polynomials Related to the Differential Equation $\frac{1}{y} \frac {dy}{dx} = \frac{a_0 + a_1x}{b_0 + b_1x + b_2x^2} \equiv \frac{N}{D}$