Heun polynomial products

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1: 31.16 Mathematical Applications

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§31.16(ii) Heun Polynomial Products

►Expansions of Heun polynomial products in terms of Jacobi polynomial (§18.3) products are derived in Kalnins and Miller (1991a, b, 1993) from the viewpoint of interrelation between two bases in a Hilbert space: … ►

2: 31.8 Solutions via Quadratures

… ►Here ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ is a polynomial of degree $g$ in $\lambda$ and of degree $N=m_0+m_1+m_2+m_3$ in $z$, that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. …

3: Bibliography W

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Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.

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External Links:

Document

Referenced by:

(20.7.34), §20.7(ix).

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Global asymptotics of the Meixner polynomials. Asymptotic Analysis 75 (3-4), pp. 211–231.

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External Links:

ISSN 0921-7134, Document, MathReview (Mario Pérez Riera), ZentralBlatt

Referenced by:

§18.24, §18.24.

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

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The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A, pp. 32–52.

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External Links:

FullText, Document, ZentralBlatt

Referenced by:

§35.3(i), §35.3(ii).

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On the central connection problem for the double confluent Heun equation. Math. Nachr. 195, pp. 267–276.

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External Links:

ISSN 0025-584X, MathReview, ZentralBlatt

Referenced by:

§31.12.

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4: Bibliography L

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The two-point connection problem for differential equations of the Heun class. Teoret. Mat. Fiz. 101 (3), pp. 360–368 (Russian).

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Notes:

In Russian. English translation: Theoret. and Math. Phys. 101(1994), no. 3, pp. 1413–1418 (1995)

External Links:

ISSN 0564-6162, Document, MathReview, ZentralBlatt

Referenced by:

§31.18.

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Asymptotic and numeric study of eigenvalues of the double confluent Heun equation. J. Phys. A 31 (42), pp. 8521–8531.

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ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§31.13, §31.18.

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The central two-point connection problem for the Heun class of ODEs. J. Phys. A 31 (18), pp. 4249–4261.

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ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§31.12.

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Heun’s equation with nearby singularities. Proc. Roy. Soc. London Ser. A 455, pp. 4347–4361.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§31.13, §31.17(ii).

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Numerical evaluation of integrals containing a spherical Bessel function by product integration. J. Math. Phys. 22 (7), pp. 1399–1413.

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ISSN 0022-2488, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.74(vii).

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5: Bibliography R

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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:

ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

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Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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External Links:

ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

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Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

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Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v).

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Heun’s Differential Equations. The Clarendon Press Oxford University Press, New York.

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External Links:

ISBN 0-19-859695-2, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§31.1, §31.12, §31.13, §31.2(v), §31.2(i), §31.2(iv), §31.3(ii), §31.4, §31.7(ii), §31.9(i), Ch.31, Profile Gerhard Wolf.

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6: Bibliography H

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Soft-core Coulomb potentials and Heun’s differential equation. J. Math. Phys. 51 (2), pp. Art. ID 022107, 19 pages.

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External Links:

ISSN 0022-2488, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§31.17(ii), §31.17(ii).

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A Table of Series and Products. Prentice-Hall, Englewood Cliffs, N.J..

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Notes:

Table erratum: Math. Comp. v.47 (1986), no. 176, p. 767.

External Links:

ZentralBlatt

Referenced by:

§10.23(iv), §10.44(iv), §10.60(iv), §10.65(iv), §11.10(x), §11.7(v), §12.13(ii), §13.11, §14.18(iv), §15.15, §18.18(ix), §24.14(iii), §25.11(xi), §25.8, §4.11, §4.27, §4.41, §5.16, §6.15, §7.15, §8.17(vi).

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Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.16.

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Orthogonal Laurent polynomials. Nederl. Akad. Wetensch. Indag. Math. 48 (1), pp. 17–36.

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External Links:

ISSN 0019-3577, Document, MathReview (W. J. Thron), ZentralBlatt

Referenced by:

§18.33(v).

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Roots of the Euler polynomials. Pacific J. Math. 64 (1), pp. 181–191.

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External Links:

MathReview (M. Marden), ZentralBlatt

Referenced by:

§24.12(ii).

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7: Errata

… ►We also discuss non-classical Laguerre polynomials and give much more details and examples on exceptional orthogonal polynomials. … ►

18.12.2 $${}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\alpha +1};\frac{(x-1)z}{2}){}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\beta +1};\frac{(x+1)z}{2})={\left(\frac{1}{2}(1-x)z\right)}^{-\frac{1}{2}\alpha }{J}_{\alpha }\left(\sqrt{2(1-x)z}\right){\left(\frac{1}{2}(1+x)z\right)}^{-\frac{1}{2}\beta }{I}_{\beta }\left(\sqrt{2(1+x)z}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{z}^n$$

This equation was updated to include on the left-hand side, its definition in terms of a product of two ${}_0{}^{}𝐅_1^{}$ functions.

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31.3.10 $$z^{-\alpha }H\mathrm{\ell }(\frac{1}{a},\frac{q}{a}-\alpha \left(\beta -ϵ\right)-\frac{\alpha }{a}\left(\beta -\delta \right);\alpha ,\alpha -\gamma +1,\alpha -\beta +1,\delta ;\frac{1}{z})$$

31.3.11 $$z^{-\beta }H\mathrm{\ell }(\frac{1}{a},\frac{q}{a}-\beta \left(\alpha -ϵ\right)-\frac{\beta }{a}\left(\alpha -\delta \right);\beta ,\beta -\gamma +1,\beta -\alpha +1,\delta ;\frac{1}{z})$$

In both equations, the second entry in the $H\mathrm{\ell }$ has been corrected with an extra minus sign.

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Some references were added to §§7.25(ii), 7.25(iii), 7.25(vi), 8.28(ii), and to ¶Products (in §10.74(vii)) and §10.77(ix).

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31.16.2

$xy=a{\sin }^2\theta {\cos }^2\varphi ,$

$(x-1)(y-1)=(1-a){\sin }^2\theta {\sin }^2\varphi ,$

$(x-a)(y-a)=a(a-1){\cos }^2\theta$

31.16.3 $$A_0=\frac{n!}{{\left(\gamma +\delta \right)}_n}{\mathrm{𝐻𝑝}}_{n,m}\left(1\right),Q_0{A}_0+R_0{A}_1=0$$

Originally $x,y$ were incorrectly defined by the set of equations (31.16.2), given previously as “$x={\sin }^2\theta {\cos }^2\varphi ,$ $y={\sin }^2\theta {\sin }^2\varphi$”. In fact, $x,y$ are implicitly defined by the corrected set of equations. In (31.16.3), the initial data $A_0$, previously missing, has now been included.

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8: Bibliography M

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On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.

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External Links:

ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§31.7(i).

Encodings:

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The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.

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External Links:

ISSN 0025-5718, Document, MathReview (Wadim Zudilin), ZentralBlatt (Masaaki Yoshida)

Referenced by:

§31.3(iii).

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On the zeros of cross-product Bessel functions. J. Math. Mech. 16, pp. 447–452.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(x).

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On the evaluation of the integral over the product of two spherical Bessel functions. J. Math. Phys. 32 (3), pp. 642–648.

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External Links:

ISSN 0022-2488, Document, MathReview (S. K. Chatterjea), ZentralBlatt

Referenced by:

§1.17(ii), §10.59.

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Analytic expressions for integrals of products of spherical Bessel functions. J. Phys. A 24 (7), pp. 1435–1453.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§10.59.

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