Szeg? theorem

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1: 18.11 Relations to Other Functions

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2: 18.15 Asymptotic Approximations

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18.15.4_5 $${\left(\sin \frac{1}{2}\theta \right)}^{\alpha +\frac{1}{2}}{\left(\cos \frac{1}{2}\theta \right)}^{\beta +\frac{1}{2}}{P}_n^{(\alpha ,\beta )}\left(\cos \theta \right)={\pi }^{-\frac{1}{2}}{n}^{-\frac{1}{2}}\cos \left(\frac{1}{2}(2n+\alpha +\beta +1)\theta -\frac{1}{4}(2\alpha +1)\pi \right)+O\left(n^{-\frac{3}{2}}\right),$$ $\alpha ,\beta \in ℝ$,

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Symbols:

$O\left(x\right)$: order not exceeding, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\in$: element of, $ℝ$: real line, $\sin z$: sine function and $n$: nonnegative integer

Proveds:

Ismail (2009, Theorem 4.3.1); Szegő (1975, Theorem 8.21.8); with attribution to Darboux(proved)

Referenced by:

§18.15(i), Erratum (V1.2.0) §18.15

Permalink:

http://dlmf.nist.gov/18.15.E4_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.15(i), §18.15 and Ch.18

… ►The first term of this expansion also appears in Szegő (1975, Theorem 8.21.7). … ►For a bound on the error term in (18.15.10) see Szegő (1975, Theorem 8.21.11). … ►Another expansion follows from (18.15.10) by taking $\lambda =\frac{1}{2}$; see Szegő (1975, Theorem 8.21.5). …

3: 18.16 Zeros

… ►For further information on the zeros of the classical orthogonal polynomials, see Szegő (1975, Chapter VI), Erdélyi et al. (1953b, §§10.16 and 10.17), Gatteschi (1987, 2002), López and Temme (1999a), and Temme (1990a). …

4: 18.33 Polynomials Orthogonal on the Unit Circle

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Szegő’s Theorem

►For $w(z)$ as in (18.33.19) (or more generally as the weight function of the absolutely continuous part of the measure $\mu$ in (18.33.17)) and with ${\alpha }_n$ the Verblunsky coefficients in (18.33.23), (18.33.24), Szegő’s theorem states that …

5: 18.2 General Orthogonal Polynomials

… ►For OP’s $p_n(x)$ with weight function in the class $𝒢$ there are asymptotic formulas as $n\to \mathrm{\infty }$, respectively for $x\in ℂ$ outside $[-1,1]$ and for $x\in [-1,1]$, see Szegő (1975, Theorems 12.1.2, 12.1.4). Under further conditions on the weight function there is an equiconvergence theorem, see Szegő (1975, Theorem 13.1.2). … ►For OP’s with orthogonality measure in $𝒮$ Nevai (1979, pp. 148–150) generalized Szegő’s equiconvergence theorem. … ►See Szegő (1975, Theorem 7.2). …

6: 18.14 Inequalities

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18.14.3_5 $${\left(\frac{1}{2}(1+x)\right)}^{\beta /2}\left|P_n^{(\alpha ,\beta )}\left(x\right)\right|\le P_n^{(\alpha ,\beta )}\left(1\right)=\frac{{\left(\alpha +1\right)}_n}{n!},$$ $-1\le x\le 1$, $\alpha ,\beta \ge 0$.

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Proved:

Carnicer et al. (2020, Theorem 2)(proved)

Proof sketch:

The case $\alpha =0$ follows from the result on monotonic weight functions in §18.2(xii).

Referenced by:

§18.14(i), §18.14(i), §18.2(xii), Erratum (V1.2.0) §18.14

Permalink:

http://dlmf.nist.gov/18.14.E3_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

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Szegő–Szász Inequality

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18.14.25 $$\sum _{m=0}^n\frac{{\left(\lambda +1\right)}_{n-m}}{(n-m)!}\frac{{\left(\lambda +1\right)}_m}{m!}\frac{P_m^{(\alpha ,\beta )}\left(x\right)}{P_m^{(\beta ,\alpha )}\left(1\right)}\ge 0,$$ $x\ge -1$, $\alpha +\beta \ge \lambda \ge 0$, $\beta \ge -\frac{1}{2}$, $n=0,1,\mathrm{\dots }$.

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Gasper (1977, Theorem 5)(proved)

Source:

Askey and Gasper (1976, Conjecture 1); conjectured

Referenced by:

Erratum (V1.2.0) §18.14

Permalink:

http://dlmf.nist.gov/18.14.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(iv), §18.14(iv), §18.14 and Ch.18

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18.14.26 $$\sum _{m=0}^n\frac{P_m^{(\alpha ,\beta )}\left(x\right)}{P_m^{(\beta ,\alpha )}\left(1\right)}\ge 0,$$ $x\ge -1$, $n=0,1,\mathrm{\dots }$,

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Gasper (1977, Theorem 4); also proved there for $\alpha +\beta \ge -2$, $\beta \ge 0$(proved)

Source:

Askey and Gasper (1976, (1.7), (1.12)); conjectured

Referenced by:

§18.14(iv), §18.38(ii)

Permalink:

http://dlmf.nist.gov/18.14.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(iv), §18.14(iv), §18.14 and Ch.18

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18.14.27 $$\sum _{m=0}^n\frac{{\left(\lambda +1\right)}_{n-m}}{(n-m)!}\frac{{\left(\lambda +1\right)}_m}{m!}\frac{{(-1)}^m{L}_m^{(\beta )}\left(x\right)}{L_m^{(\beta )}\left(0\right)}\ge 0,$$ $x\ge 0$,  $\beta ,\lambda \ge -\frac{1}{2}$,  $n=0,1,\mathrm{\dots }$.

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Gasper (1977, Theorem 1)(proved)

Source:

Askey and Gasper (1976, Conjecture 2); conjectured

Referenced by:

Erratum (V1.2.0) §18.14

Permalink:

http://dlmf.nist.gov/18.14.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(iv), §18.14(iv), §18.14 and Ch.18

7: Bibliography S

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Szegő’s Theorem and Its Descendants. Spectral Theory for $L^2$ Perturbations of Orthogonal Polynomials. M. B. Porter Lectures, Princeton University Press, Princeton, NJ.

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External Links:

ISBN 978-0-691-14704-8, MathReview (Harry Dym)

Referenced by:

§18.2(xi).

Encodings:

BibTeX

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8: 18.18 Sums

… ►See Szegő (1975, Theorems 3.1.5 and 5.7.1). …

9: 18.17 Integrals

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10: Errata

… ►In regard to orthogonal polynomials on the unit circle, we now discuss monic polynomials, Verblunsky’s Theorem, and Szegő’s theorem. …