Euler–Poisson–Darboux equation

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21: 18.12 Generating Functions

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

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (9.8.12))

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) for Equation (18.12.2)

Permalink:

http://dlmf.nist.gov/18.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.2_5 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\gamma ,\alpha +\beta +1-\gamma }{\alpha +1};\frac{1-R-z}{2}){}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\gamma ,\alpha +\beta +1-\gamma }{\beta +1};\frac{1-R+z}{2})=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\gamma \right)}_n{\left(\alpha +\beta +1-\gamma \right)}_n}{{\left(\alpha +1\right)}_n{\left(\beta +1\right)}_n}{P}_n^{(\alpha ,\beta )}\left(x\right)z^n,$$ $R=\sqrt{1-2xz+z^2}$, ,

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Proved:

Rainville (1960, §141, (2)); Brafman’s generating function(proved)

Source:

Koekoek et al. (2010, (9.8.15))

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) Section 18.12

Permalink:

http://dlmf.nist.gov/18.12.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.12, §18.12 and Ch.18

►with $\gamma$ arbitrary. Note that (18.12.2_5) yields (18.12.1) by putting $\gamma =0$ and (18.12.2) by replacing $z$ by $-{\gamma }^{-2}z$ and next letting $\gamma \to \mathrm{\infty }$. … ►See §18.18(vii) for Poisson kernels; these are special cases of bilateral generating functions.

22: 24.4 Basic Properties

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§24.4(i) Difference Equations

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§24.4(ii) Symmetry

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§24.4(iii) Sums of Powers

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§24.4(iv) Finite Expansions

… ►Next, …

23: 31.14 General Fuchsian Equation

§31.14 General Fuchsian Equation

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§31.14(i) Definitions

… ►The exponents at the finite singularities $a_j$ are $\{0,1-{\gamma }_j\}$ and those at $\mathrm{\infty }$ are $\{\alpha ,\beta \}$, where …The three sets of parameters comprise the singularity parameters $a_j$, the exponent parameters $\alpha ,\beta ,{\gamma }_j$, and the $N-2$ free accessory parameters $q_j$. … ►

Normal Form

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24: 30.6 Functions of Complex Argument

… ►The solutions …of (30.2.1) with $\mu =m$ and $\lambda ={\lambda }_n^m\left({\gamma }^2\right)$ are real when $z\in (1,\mathrm{\infty })$, and their principal values (§4.2(i)) are obtained by analytic continuation to $ℂ\setminus (-\mathrm{\infty },1]$. … ►with $A_n^{\pm m}({\gamma }^2)$ as in (30.11.4). … ►For results for Equation (30.2.1) with complex parameters see Meixner and Schäfke (1954).

25: 1.8 Fourier Series

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1.8.5 $$\frac{1}{\pi }{\int }_{-\pi }^{\pi }{\left|f(x)\right|}^{2}dx=\frac{1}{2}{\left|a_0\right|}^2+\sum _{n=1}^{\mathrm{\infty }}({\left|a_n\right|}^2+{\left|b_n\right|}^2),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

Erratum (V1.2.0) for Equations (1.8.5), (1.8.6)

Permalink:

http://dlmf.nist.gov/1.8.E5

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

Previously this equation was given as an inequality. For square integrable functions this inequality $\ge$ can be sharpened to $=$.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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1.8.6 $$\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{\left|f(x)\right|}^{2}dx=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\left|c_n\right|}^2,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.8(i), Erratum (V1.2.0) for Equations (1.8.5), (1.8.6)

Permalink:

http://dlmf.nist.gov/1.8.E6

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

Previously this equation was given as an inequality. For square integrable functions this inequality $\ge$ can be sharpened to $=$.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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§1.8(iv) Poisson’s Summation Formula

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1.8.13Moved to (1.8.6_1).

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Referenced by:

(1.8.6_1), §1.8(i), §1.8(iv)

Permalink:

http://dlmf.nist.gov/1.8.E13

Rearrangement (effective with 1.2.0):

This equation has been moved to (1.8.6_1).

See also:

Annotations for §1.8(iv), §1.8 and Ch.1

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1.8.16 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-{(n+x)}^2\omega }=\sqrt{\frac{\pi }{\omega }}\left(1+2\sum _{n=1}^{\mathrm{\infty }}{\mathrm{e}}^{-n^2{\pi }^2/\omega }\cos \left(2n\pi x\right)\right),$$ $\mathrm{\Re }\omega >0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $n$: nonnegative integer

Referenced by:

§1.8(iv)

Permalink:

http://dlmf.nist.gov/1.8.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(iv), §1.8 and Ch.1

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26: 16.16 Transformations of Variables

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16.16.5 $$F_3(\alpha ,\gamma -\alpha ;\beta ,\gamma -\beta ;\gamma ;x,y)={(1-y)}^{\alpha +\beta -\gamma }{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha ,\beta }{\gamma };x+y-xy),$$

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Symbols:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function and ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function

Permalink:

http://dlmf.nist.gov/16.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

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16.16.5_5 $$F_4(\alpha ,\beta ;\gamma ,\beta ;x(1-y),y(1-x))={(1-x)}^{-\alpha }{(1-y)}^{-\alpha }{F}_1(\alpha ;\gamma -\beta ,\alpha -\gamma +1;\gamma ;\frac{x}{x-1},\frac{xy}{(1-x)(1-y)}),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function and $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function

Referenced by:

§16.16(i), Erratum (V1.1.12) for Additions

Permalink:

http://dlmf.nist.gov/16.16.E5_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.12):

This equation was added.

See also:

Annotations for §16.16(i), §16.16 and Ch.16

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16.16.7 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x(1-y),y(1-x))=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_k{\left(\beta \right)}_k{\left(\alpha +\beta -\gamma -{\gamma }^{\prime }+1\right)}_k}{{\left(\gamma \right)}_k{\left({\gamma }^{\prime }\right)}_{k}k!}{x}^k{y}^k{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha +k,\beta +k}{\gamma +k};x){}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha +k,\beta +k}{{\gamma }^{\prime }+k};y);$$

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Symbols:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

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16.16.9 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)={(1-x)}^{-\alpha }{F}_2(\alpha ;\gamma -\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };\frac{x}{x-1},\frac{y}{1-x}),$$

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Symbols:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function

Permalink:

http://dlmf.nist.gov/16.16.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(ii), §16.16 and Ch.16

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16.16.10 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)=\frac{\mathrm{\Gamma }\left({\gamma }^{\prime }\right)\mathrm{\Gamma }\left(\beta -\alpha \right)}{\mathrm{\Gamma }\left({\gamma }^{\prime }-\alpha \right)\mathrm{\Gamma }\left(\beta \right)}{(-y)}^{-\alpha }{F}_4(\alpha ,\alpha -{\gamma }^{\prime }+1;\gamma ,\alpha -\beta +1;\frac{x}{y},\frac{1}{y})+\frac{\mathrm{\Gamma }\left({\gamma }^{\prime }\right)\mathrm{\Gamma }\left(\alpha -\beta \right)}{\mathrm{\Gamma }\left({\gamma }^{\prime }-\beta \right)\mathrm{\Gamma }\left(\alpha \right)}{(-y)}^{-\beta }{F}_4(\beta ,\beta -{\gamma }^{\prime }+1;\gamma ,\beta -\alpha +1;\frac{x}{y},\frac{1}{y}).$$

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Symbols:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function and $\mathrm{\Gamma }\left(z\right)$: gamma function

Permalink:

http://dlmf.nist.gov/16.16.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(ii), §16.16 and Ch.16

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27: Bibliography W

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Prime Divisors of the Bernoulli and Euler Numbers. In Number Theory for the Millennium, III (Urbana, IL, 2000), pp. 357–374.

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External Links:

FullText, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.20.

Encodings:

BibTeX

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The zeros of Euler’s psi function and its derivatives. J. Math. Anal. Appl. 332 (1), pp. 607–616.

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External Links:

ISSN 0022-247X, Document, MathReview, ZentralBlatt (Mehdi Hassani)

Referenced by:

§5.4(iii).

Encodings:

BibTeX

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Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.

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External Links:

ISSN 0025-5718, Document, MathReview (B. G. Pachpatte), ZentralBlatt

Referenced by:

§2.9(iii).

Encodings:

BibTeX

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The method of Darboux. J. Approximation Theory 10 (2), pp. 159–171.

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External Links:

Document, MathReview (M. E. Noble), ZentralBlatt

Referenced by:

§2.10(iv).

Encodings:

BibTeX

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On a uniform treatment of Darboux’s method. Constr. Approx. 21 (2), pp. 225–255.

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External Links:

ISSN 0176-4276, Document, MathReview Entry, ZentralBlatt

Referenced by:

§2.10(iv).

Encodings:

BibTeX

…

28: 15.11 Riemann’s Differential Equation

§15.11 Riemann’s Differential Equation

… ►The most general form is given by … ►Here $\{a_1,a_2\}$, $\{b_1,b_2\}$, $\{c_1,c_2\}$ are the exponent pairs at the points $\alpha$, $\beta$, $\gamma$, respectively. …Also, if any of $\alpha$, $\beta$, $\gamma$, is at infinity, then we take the corresponding limit in (15.11.1). … ►These constants can be chosen to map any two sets of three distinct points $\{\alpha ,\beta ,\gamma \}$ and $\{\tilde{\alpha },\tilde{\beta },\tilde{\gamma }\}$ onto each other. …

29: 20.11 Generalizations and Analogs

… ►This is the discrete analog of the Poisson identity (§1.8(iv)). … ►The first of equations (20.9.2) can also be written … ►The importance of these combined theta functions is that sets of twelve equations for the theta functions often can be replaced by corresponding sets of three equations of the combined theta functions, plus permutation symmetry. Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …

30: 8.2 Definitions and Basic Properties

… ►The general values of the incomplete gamma functions $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ are defined by … ►In this subsection the functions $\gamma$ and $\mathrm{\Gamma }$ have their general values. ►The function ${\gamma }^{\ast }(a,z)$ is entire in $z$ and $a$. … ►If $w=\gamma (a,z)$ or $\mathrm{\Gamma }(a,z)$, then … ►If $w={\mathrm{e}}^z{z}^{1-a}\mathrm{\Gamma }(a,z)$, then …