Rogers%E2%80%93Szeg%C5%91%20polynomials

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21: 18.28 Askey–Wilson Class

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Duality

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§18.28(v) Continuous $q$-Ultraspherical Polynomials

… ►These polynomials are also called Rogers polynomials. ►

§18.28(vi) Continuous $q$-Hermite Polynomials

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§18.28(viii) $q$-Racah Polynomials

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22: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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Rogers–Fine Identity

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23: Bibliography D

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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.

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ISSN 0022-314X, Document, MathReview (T. Metsänkylä)

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§24.12(iii).

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item.

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Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).

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Notes:

With an appendix by I. M. Krichever. In Russian. English translation: Russian Math. Surveys 36(1981), no. 2, pp. 11–92 (1982)

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ISSN 0042-1316, Document, MathReview (Alexander A. Pankov), ZentralBlatt

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§21.1, §21.6(ii), §21.7(i), Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2, Ch.21, Notations T.

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Error analysis in a uniform asymptotic expansion for the generalised exponential integral. J. Comput. Appl. Math. 80 (1), pp. 127–161.

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ISSN 0377-0427, Document, MathReview (Eberhard Wagner), ZentralBlatt

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§2.11(iii), §8.20(ii).

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Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.

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ISSN 0021-9045, Document, MathReview (Eberhard Wagner), ZentralBlatt

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§18.24.

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24: 8.23 Statistical Applications

… ►In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of $Q(a,x)$; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …

25: 32.8 Rational Solutions

… ►where the $Q_n(z)$ are monic polynomials (coefficient of highest power of $z$ is $1$) satisfying … ►Next, let $p_m(z)$ be the polynomials defined by $p_m(z)=0$ for , and … ►where $P_m(z)$ and $Q_m(z)$ are polynomials of degree $m$, with no common zeros. … ►where $P_{j,n-1}(z)$ and $Q_{j,n}(z)$ are polynomials of degrees $n-1$ and $n$, respectively, with no common zeros. … ►where $\lambda$, $\mu$ are constants, and $P_{n-1}(z)$, $Q_n(z)$ are polynomials of degrees $n-1$ and $n$, respectively, with no common zeros. …

26: Bibliography K

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A proof of the $q$-Macdonald-Morris conjecture for $B{C}_n$ . Mem. Amer. Math. Soc. 108 (516), pp. vi+80.

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ISSN 0065-9266, MathReview (Eric M. Opdam), ZentralBlatt

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§17.14.

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Linear convergence and the bisection algorithm. Amer. Math. Monthly 93 (1), pp. 48–51.

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ISSN 0002-9890, FullText, MathReview (J. G. Herriot), ZentralBlatt

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§3.8(iii).

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

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1st item.

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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Document, MathReview (Matti Jutila), ZentralBlatt

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§25.18(i).

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

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§32.11(iv).

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27: 26.13 Permutations: Cycle Notation

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28: 26.6 Other Lattice Path Numbers

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Table 26.6.1: Delannoy numbers $D(m,n)$.

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$m$	$n$
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10	1	21	221	1561	8361	36365	1 34245	4 33905	12 56465	33 17445	80 97453

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Table 26.6.3: Narayana numbers $N(n,k)$.

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$n$	$k$
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5	0	1	10	20	10	1
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29: Bibliography J

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Fonctions de Mathieu et polynômes de Klein-Gordon. C. R. Acad. Sci. Paris Sér. I Math. 325 (7), pp. 713–716 (French).

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Notes:

Translated title: Mathieu functions and Klein-Gordon polynomials.

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ISSN 0764-4442, Document, MathReview, ZentralBlatt

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7th item.

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Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Phys. D 1 (1), pp. 80–158.

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ISSN 0167-2789, Document, MathReview, ZentralBlatt

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§32.16.

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Uniform asymptotic expansions for Meixner polynomials. Constr. Approx. 14 (1), pp. 113–150.

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ISSN 0176-4276, Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§18.24, §2.4(vi).

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Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.

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ISSN 0021-9045, Document, MathReview (N. Hayek Calil), ZentralBlatt

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§18.24.

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Modifications of Coulombic interactions by polarizable atoms. Math. Proc. Cambridge Philos. Soc. 80 (3), pp. 535–539.

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ISSN 0305-0041, Document, MathReview (Cataldo Agostinelli)

Referenced by:

§34.12.

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30: 22.7 Landen Transformations

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