Heine transformations (first, second, third)

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1: 1.14 Integral Transforms

§1.14 Integral Transforms

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§1.14(i) Fourier Transform

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§1.14(iii) Laplace Transform

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Fourier Transform

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Laplace Transform

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2: 14.28 Sums

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14.28.1 $$P_{\nu }\left(z_1{z}_2-{\left(z_1^2-1\right)}^{1/2}{\left(z_2^2-1\right)}^{1/2}\cos \varphi \right)=P_{\nu }\left(z_1\right)P_{\nu }\left(z_2\right)+2\sum _{m=1}^{\mathrm{\infty }}{(-1)}^m\frac{\mathrm{\Gamma }\left(\nu -m+1\right)}{\mathrm{\Gamma }\left(\nu +m+1\right)}{P}_{\nu }^m\left(z_1\right)P_{\nu }^m(z_2)\cos \left(m\varphi \right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\cos z$: cosine function, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $z$: complex variable, $m$: nonnegative integer and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.28(i), §14.28 and Ch.14

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§14.28(ii) Heine’s Formula

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14.28.2 $$\sum _{n=0}^{\mathrm{\infty }}(2n+1)Q_n\left(z_1\right)P_n\left(z_2\right)=\frac{1}{z_1-z_2},$$ $z_1\in {ℰ}_1$, $z_2\in {ℰ}_2$,

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Symbols:

$\in$: element of, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $z$: complex variable, $n$: nonnegative integer and ${ℰ}_1$, ${ℰ}_2$: ellipses

Permalink:

http://dlmf.nist.gov/14.28.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.28(ii), §14.28 and Ch.14

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3: 18.7 Interrelations and Limit Relations

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§18.7(i) Linear Transformations

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§18.7(ii) Quadratic Transformations

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18.7.17 $$U_{2n}\left(x\right)=W_n\left(2x^2-1\right),$$

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Symbols:

$W_n\left(x\right)$: Chebyshev polynomial of the fourth kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $V_n\left(x\right)$: Chebyshev polynomial of the third kind, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.30

Referenced by:

§18.7(ii), Erratum (V1.0.28) for Table 18.3.1

Permalink:

http://dlmf.nist.gov/18.7.E17

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_n\left(x\right)$, $W_n\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_n\left(x\right)$, $W_n\left(x\right)$, having been interchanged, on the right-hand side we replaced $V_n\left(2x^2-1\right)$, with $W_n\left(2x^2-1\right)$. For further details see Errata.

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

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18.7.18 $$T_{2n+1}\left(x\right)=x{V}_n\left(2x^2-1\right).$$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $W_n\left(x\right)$: Chebyshev polynomial of the fourth kind, $V_n\left(x\right)$: Chebyshev polynomial of the third kind, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.29

Referenced by:

§18.7(ii), Erratum (V1.0.28) for Table 18.3.1

Permalink:

http://dlmf.nist.gov/18.7.E18

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_n\left(x\right)$, $W_n\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_n\left(x\right)$, $W_n\left(x\right)$, having been interchanged, on the right-hand side we replaced $x{W}_n\left(2x^2-1\right)$, with $x{V}_n\left(2x^2-1\right)$. For further details see Errata.

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

… ► See §18.11(ii) for limit formulas of Mehler–Heine type.

4: Bibliography V

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A Fortran computer program for calculating the oblate spheroidal radial functions of the first and second kind and their first derivatives. NRL Report No. 6959 Naval Res. Lab.  Washingtion, D.C..

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Referenced by:

§30.18(iii).

Encodings:

BibTeX

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Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives. Quart. Appl. Math. 62 (3), pp. 493–507.

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Notes:

For software, see Van Buren (website)

External Links:

ISSN 0033-569X, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§30.16(iii), §30.16(iii).

Encodings:

BibTeX

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On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.12, §19.5.

Encodings:

BibTeX

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Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.

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External Links:

ISSN 0377-0427, Document, Link, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§15.8(v).

Encodings:

BibTeX

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Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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External Links:

ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt

Referenced by:

§31.15(iii).

Encodings:

BibTeX

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5: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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Heine’s First Transformation

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Heine’s Second Tranformation

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Heine’s Third Transformation

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Fine’s First Transformation

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Heine’s Contiguous Relations

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6: 10.1 Special Notation

… ►The main functions treated in this chapter are the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$; Hankel functions $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$; modified Bessel functions $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$; spherical Bessel functions ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$; modified spherical Bessel functions ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, ${𝗄}_n\left(z\right)$; Kelvin functions ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. … ►Abramowitz and Stegun (1964): $j_n(z)$, $y_n(z)$, $h_n^{(1)}(z)$, $h_n^{(2)}(z)$, for ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$, respectively, when $n\ge 0$. ►Jeffreys and Jeffreys (1956): ${\mathrm{Hs}}_{\nu }(z)$ for $H_{\nu }^{(1)}\left(z\right)$, ${\mathrm{Hi}}_{\nu }(z)$ for $H_{\nu }^{(2)}\left(z\right)$, ${\mathrm{Kh}}_{\nu }(z)$ for $(2/\pi )K_{\nu }\left(z\right)$. ►Whittaker and Watson (1927): $K_{\nu }\left(z\right)$ for $\cos \left(\nu \pi \right)K_{\nu }\left(z\right)$. ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

7: 14.12 Integral Representations

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14.12.2 $${𝖯}_{\nu }^{-\mu }\left(x\right)=\frac{{\left(1-x^2\right)}^{-\mu /2}}{\mathrm{\Gamma }\left(\mu \right)}{\int }_x^1{𝖯}_{\nu }\left(t\right){(t-x)}^{\mu -1}dt,$$ $\mathrm{\Re }\mu >0$;

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.17(i)

Permalink:

http://dlmf.nist.gov/14.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(i), §14.12(i), §14.12 and Ch.14

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14.12.3 $${𝖰}_{\nu }^{\mu }\left(\cos \theta \right)=\frac{{\pi }^{1/2}\mathrm{\Gamma }\left(\nu +\mu +1\right){(\sin \theta )}^{\mu }}{2^{\mu +1}\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}\left({\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{(\cos \theta +\mathrm{i}\sin \theta \cosh t)}^{\nu +\mu +1}}dt+{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{(\cos \theta -\mathrm{i}\sin \theta \cosh t)}^{\nu +\mu +1}}dt\right),$$ , $\mathrm{\Re }\mu >-\frac{1}{2}$, $\mathrm{\Re }\nu \pm \mu >-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(i), §14.12(i), §14.12 and Ch.14

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14.12.5 $$P_{\nu }^{-\mu }\left(x\right)=\frac{{\left(x^2-1\right)}^{-\mu /2}}{\mathrm{\Gamma }\left(\mu \right)}{\int }_1^x{P}_{\nu }\left(t\right){(x-t)}^{\mu -1}dt,$$ $\mathrm{\Re }\mu >0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.17(i)

Permalink:

http://dlmf.nist.gov/14.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12 and Ch.14

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14.12.12 $${𝑸}_n^m\left(x\right)=\frac{1}{(n-m)!}{P}_n^m\left(x\right){\int }_x^{\mathrm{\infty }}\frac{dt}{\left(t^2-1\right){\left(P_n^m\left(t\right)\right)}^2},$$ $n\ge m$.

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§14.17(i)

Permalink:

http://dlmf.nist.gov/14.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12 and Ch.14

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Heine’s Integral

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8: 18.11 Relations to Other Functions

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18.11.1 $${𝖯}_n^m\left(x\right)={\left(\frac{1}{2}\right)}_m{(-2)}^m{(1-x^2)}^{\frac{1}{2}m}{C}_{n-m}^{(m+\frac{1}{2})}\left(x\right)={\left(n+1\right)}_m{(-2)}^{-m}{(1-x^2)}^{\frac{1}{2}m}{P}_{n-m}^{(m,m)}\left(x\right),$$ $0\le m\le n$.

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Symbols:

: Ferrers function of the first kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§14.3(iv), §18.11(i)

Permalink:

http://dlmf.nist.gov/18.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

►For the Ferrers function ${𝖯}_n^m\left(x\right)$, see §14.3(i). … ►

§18.11(ii) Formulas of Mehler–Heine Type

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18.11.5 $$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{\alpha }}{P}_n^{(\alpha ,\beta )}\left(1-\frac{z^2}{2n^2}\right)=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{\alpha }}{P}_n^{(\alpha ,\beta )}\left(\cos \frac{z}{n}\right)=\frac{2^{\alpha }}{z^{\alpha }}{J}_{\alpha }\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

22.15.1

Referenced by:

§18.11(ii), §18.11(ii)

Permalink:

http://dlmf.nist.gov/18.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18

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18.11.6 $$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{\alpha }}{L}_n^{(\alpha )}\left(\frac{z}{n}\right)=\frac{1}{z^{\frac{1}{2}\alpha }}{J}_{\alpha }\left(2z^{\frac{1}{2}}\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

22.15.2

Referenced by:

§18.11(ii)

Permalink:

http://dlmf.nist.gov/18.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18

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9: 19.7 Connection Formulas

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§19.7(i) Complete Integrals of the First and Second Kinds

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Reciprocal-Modulus Transformation

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Imaginary-Modulus Transformation

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Imaginary-Argument Transformation

… ►For two further transformations of this type see Erdélyi et al. (1953b, p. 316). …

10: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

§17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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F. H. Jackson’s Transformations

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Transformations of ${}_3{}^{}\varphi _2^{}$-Series

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Sears–Carlitz Transformation

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Mixed-Base Heine-Type Transformations

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