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31: 31.1 Special Notation

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$x$, $y$	real variables.
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$a$	complex parameter, $|a|\ge 1,a\ne 1$.
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32: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

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28.25.1 $${\mathrm{M}}_{\nu }^{(3,4)}(z,h)\sim \frac{{\mathrm{e}}^{\pm \mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi \right)}}{{\left(\pi h(\cosh z+1)\right)}^{\frac{1}{2}}}\sum _{m=0}^{\mathrm{\infty }}\frac{D_m^{\pm }}{{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)}^m},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.1 (for 3 and for integer $\nu$) 20.9.3 (for 4 and for integer $\nu$)

Referenced by:

§28.25

Permalink:

http://dlmf.nist.gov/28.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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28.25.3 $$(m+1)D_{m+1}^{\pm }+\left({(m+\frac{1}{2})}^2\pm (m+\frac{1}{4})8\mathrm{i}h+2h^2-a\right)D_m^{\pm }\pm (m-\frac{1}{2})\left(8\mathrm{i}hm\right)D_{m-1}^{\pm }=0,$$ $m\ge 0$.

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Symbols:

$\mathrm{i}$: imaginary unit, $m$: integer, $h$: parameter, $a$: parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.2 (for $+$) 20.9.4 (for $-$)

Permalink:

http://dlmf.nist.gov/28.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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28.25.4 $$\mathrm{\Re }z\to +\mathrm{\infty },$$ $-\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le 2\pi -\delta$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $h$: parameter, $z$: complex variable and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/28.25.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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28.25.5 $$\mathrm{\Re }z\to +\mathrm{\infty },$$ $-2\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le \pi -\delta$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $h$: parameter, $z$: complex variable and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/28.25.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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33: 28.4 Fourier Series

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28.4.19 $$A_{2m+1}^{2n+1}(-q)={(-1)}^{n-m}{B}_{2m+1}^{2n+1}(q),$$

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Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer, $A_m(q)$: Fourier coefficient and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(v), §28.4 and Ch.28

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28.4.20 $$B_{2m+1}^{2n+1}(-q)={(-1)}^{n-m}{A}_{2m+1}^{2n+1}(q).$$

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Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer, $A_m(q)$: Fourier coefficient and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(v), §28.4 and Ch.28

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28.4.21 $$A_{2s}^0(q)=\left(\frac{{(-1)}^{s}2}{{(s!)}^2}{\left(\frac{q}{4}\right)}^s+O\left(q^{s+2}\right)\right)A_0^0(q),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), $q=h^2$: parameter and $A_m(q)$: Fourier coefficient

A&S Ref:

20.2.29 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.4.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(vi), §28.4 and Ch.28

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28.4.22 $$\begin{array}{c}{A}_{m+2s}^m(q)\\ B_{m+2s}^m(q)\end{array}\}=\left(\frac{{(-1)}^{s}m!}{s!(m+s)!}{\left(\frac{q}{4}\right)}^s+O\left(q^{s+1}\right)\right)\{\begin{array}{c}{A}_m^m(q),\\ B_m^m(q),\end{array}$$

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Symbols:

$O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $A_m(q)$: Fourier coefficient and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(vi), §28.4 and Ch.28

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28.4.23 $$\begin{array}{c}{A}_{m-2s}^m(q)\\ B_{m-2s}^m(q)\end{array}\}=\left(\frac{(m-s-1)!}{s!(m-1)!}{\left(\frac{q}{4}\right)}^s+O\left(q^{s+1}\right)\right)\{\begin{array}{c}{A}_m^m(q),\\ B_m^m(q).\end{array}$$

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Symbols:

$O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $A_m(q)$: Fourier coefficient and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(vi), §28.4 and Ch.28

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34: 28.15 Expansions for Small $q$

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28.15.2 $$a-{\nu }^2-\frac{q^2}{a-{(\nu +2)}^2-}\frac{q^2}{a-{(\nu +4)}^2-}\mathrm{\cdots }=\frac{q^2}{a-{(\nu -2)}^2-}\frac{q^2}{a-{(\nu -4)}^2-}\mathrm{\cdots }.$$

ⓘ

Symbols:

$q=h^2$: parameter, $\nu$: complex parameter and $a$: parameter

Referenced by:

item (e)

Permalink:

http://dlmf.nist.gov/28.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

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35: 24.17 Mathematical Applications

… ►Let $0\le h\le 1$ and $a,m$, and $n$ be integers such that $n>a$, $m>0$, and $f^{(m)}(x)$ is absolutely integrable over $[a,n]$. … ►

24.17.1 $$\sum _{j=a}^{n-1}{(-1)}^{j}f(j+h)=\frac{1}{2}\sum _{k=0}^{m-1}\frac{E_k\left(h\right)}{k!}\left({(-1)}^{n-1}{f}^{(k)}(n)+{(-1)}^a{f}^{(k)}(a)\right)+R_m(n),$$

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $!$: factorial (as in $n!$), $j$: integer, $k$: integer, $m$: integer, $h$: parameter, $a$: integer, $n$: integer, $f^{(m)}(x)$: integrable function and $R_m(n)$: remainder

Permalink:

http://dlmf.nist.gov/24.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(i), §24.17(i), §24.17 and Ch.24

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24.17.2 $$R_m(n)=\frac{1}{2(m-1)!}{\int }_a^n{f}^{(m)}(x){\tilde{E}}_{m-1}\left(h-x\right)dx.$$

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Symbols:

$dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, ${\tilde{E}}_n\left(x\right)$: periodic Euler functions, $m$: integer, $x$: real or complex, $h$: parameter, $a$: integer, $n$: integer, $f^{(m)}(x)$: integrable function and $R_m(n)$: remainder

Permalink:

http://dlmf.nist.gov/24.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(i), §24.17(i), §24.17 and Ch.24

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36: Tom H. Koornwinder

… ►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. …

37: 11.7 Integrals and Sums

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11.7.13 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{𝐇}_0\left(t\right)dt=\frac{2}{\pi \sqrt{1+a^2}}\ln \left(\frac{1+\sqrt{1+a^2}}{a}\right),$$

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $a$: parameter

Permalink:

http://dlmf.nist.gov/11.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(iii), §11.7 and Ch.11

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11.7.14 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{𝐇}_1\left(t\right)dt=\frac{2}{\pi a}-\frac{2a}{\pi \sqrt{1+a^2}}\ln \left(\frac{1+\sqrt{1+a^2}}{a}\right),$$

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $a$: parameter

Permalink:

http://dlmf.nist.gov/11.7.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(iii), §11.7 and Ch.11

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11.7.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{𝐋}_0\left(t\right)dt=\frac{2}{\pi \sqrt{a^2-1}}\arcsin \left(\frac{1}{a}\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arcsin z$: arcsine function, ${𝐋}_{\nu }\left(z\right)$: modified Struve function and $a$: parameter

Permalink:

http://dlmf.nist.gov/11.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(iii), §11.7 and Ch.11

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38: 15.1 Special Notation

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$x$	real variable.
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$a,b,c$	real or complex parameters.
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15.1.1 $${}_2{}^{}F_1^{}(a,b;c;z)=F(a,b;c;z)=F(\genfrac{}{}{0pt}{}{a,b}{c};z),$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.1.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.1 and Ch.15

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15.1.2 $$\frac{F(a,b;c;z)}{\mathrm{\Gamma }\left(c\right)}=𝐅(a,b;c;z)=𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)={}_2{}^{}𝐅_1^{}(a,b;c;z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.1.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.1 and Ch.15

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39: 33.19 Power-Series Expansions in $r$

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33.19.3 $$2\pi h(ϵ,\mathrm{\ell };r)=\sum _{k=0}^{2\mathrm{\ell }}\frac{(2\mathrm{\ell }-k)!{\gamma }_k}{k!}{(2r)}^{k-\mathrm{\ell }}-\sum _{k=0}^{\mathrm{\infty }}{\delta }_k{r}^{k+\mathrm{\ell }+1}-A(ϵ,\mathrm{\ell })\left(2\ln |2r/\kappa |+\mathrm{\Re }\psi \left(\mathrm{\ell }+1+\kappa \right)+\mathrm{\Re }\psi \left(-\mathrm{\ell }+\kappa \right)\right)f(ϵ,\mathrm{\ell };r),$$ $r\ne 0$.

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $\kappa$: quantity, $A(ϵ,\mathrm{\ell })$: function, ${\gamma }_k$: term and ${\delta }_k$: term

Referenced by:

§33.19

Permalink:

http://dlmf.nist.gov/33.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

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33.19.6 $$k(k+2\mathrm{\ell }+1){\delta }_k+2{\delta }_{k-1}+ϵ{\delta }_{k-2}+2(2k+2\mathrm{\ell }+1)A(ϵ,\mathrm{\ell }){\alpha }_k=0,$$ $k=2,3,\mathrm{\dots }$,

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Symbols:

$k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function, ${\alpha }_k$: term and ${\delta }_k$: term

Permalink:

http://dlmf.nist.gov/33.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

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40: 31.15 Stieltjes Polynomials

… ►If $z_1,z_2,\mathrm{\dots },z_n$ are the zeros of an $n$th degree Stieltjes polynomial $S(z)$, then every zero $z_k$ is either one of the parameters $a_j$ or a solution of the system of equations ►

31.15.2 $$\sum _{j=1}^N\frac{{\gamma }_j/2}{z_k-a_j}+\sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^n\frac{1}{z_k-z_j}=0,$$ $k=1,2,\mathrm{\dots },n$.

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(ii)

Permalink:

http://dlmf.nist.gov/31.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

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31.15.6 $j=1,2,\mathrm{\dots },N-1$,

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Symbols:

$j$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(iii)

Permalink:

http://dlmf.nist.gov/31.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

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31.15.7 $$q_j={\gamma }_j\sum _{k=1}^n\frac{1}{z_k-a_j},$$ $j=1,2,\mathrm{\dots },N$.

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

… ►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index $𝐦=(m_1,m_2,\mathrm{\dots },m_{N-1})$, where each $m_j$ is a nonnegative integer, there is a unique Stieltjes polynomial with $m_j$ zeros in the open interval $(a_j,a_{j+1})$ for each $j=1,2,\mathrm{\dots },N-1$. …