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31: 24.5 Recurrence Relations

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24.5.1 $$\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k\left(x\right)=n{x}^{n-1},$$ $n=2,3,\mathrm{\dots }$,

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(i), §24.5 and Ch.24

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$a_n={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\displaystyle \frac{b_{n-k}}{k+1}},$

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$b_n={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)B_k{a}_{n-k}.$

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$a_n={\displaystyle \sum _{k=0}^{\lfloor n/2\rfloor }}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2k}}\right)b_{n-2k},$

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$b_n={\displaystyle \sum _{k=0}^{\lfloor n/2\rfloor }}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2k}}\right)E_{2k}{a}_{n-2k}.$

32: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►where the infinite sum means convergence in norm, … ►The sum of the kinetic and potential energies give the quantum Hamiltonian, or energy operator; often also referred to as a Schrödinger operator. … ►Should an eigenvalue correspond to more than a single linearly independent eigenfunction, namely a multiplicity greater than one, all such eigenfunctions will always be implied as being part of any sums or integrals over the spectrum. … ►Spectral expansions of $T$, and of functions $F(T)$ of $T$, these being expansions of $T$ and $F(T)$ in terms of the eigenvalues and eigenfunctions summed over the spectrum, then follow: … ►In what follows, integrals over the continuous parts of the spectrum will be denoted by ${𝝈}_c$, and sums over the discrete spectrum by ${𝝈}_p$, with $𝝈={𝝈}_c\cup {𝝈}_p$ denoting the full spectrum. …

33: 10.60 Sums

§10.60 Sums

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§10.60(i) Addition Theorems

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§10.60(ii) Duplication Formulas

… ►For further sums of series of spherical Bessel functions, or modified spherical Bessel functions, see §6.10(ii), Luke (1969b, pp. 55–58), Vavreck and Thompson (1984), Harris (2000), and Rottbrand (2000). ►

§10.60(iv) Compendia

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34: 26.10 Integer Partitions: Other Restrictions

… ►where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … ►where the inner sum is the sum of all positive odd divisors of $t$. … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $k$ for which $n-(3k^2\pm k)\ge 0$. … ►where the inner sum is the sum of all positive divisors of $t$ that are in $S$. …

35: 30.4 Functions of the First Kind

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30.4.4 $${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)={(1-x^2)}^{\frac{1}{2}m}\sum _{k=0}^{\mathrm{\infty }}{g}_k{x}^k,$$ $-1\le x\le 1$,

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Symbols:

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

A&S Ref:

21.7.18

Permalink:

http://dlmf.nist.gov/30.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(iii), §30.4 and Ch.30

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30.4.7 $$f(x)=\sum _{n=m}^{\mathrm{\infty }}{c}_n{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),$$

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Symbols:

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $f(x)$: function and $c_n$: coefficient

Referenced by:

§30.4(iv)

Permalink:

http://dlmf.nist.gov/30.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(iv), §30.4 and Ch.30

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30.4.9 $$\underset{N\to \mathrm{\infty }}{lim}{\int }_{-1}^1{\left|f(x)-\sum _{n=m}^N{c}_n{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)\right|}^{2}dx=0.$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $f(x)$: function and $c_n$: coefficient

Permalink:

http://dlmf.nist.gov/30.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(iv), §30.4 and Ch.30

►It is also equiconvergent with its expansion in Ferrers functions (as in (30.4.2)), that is, the difference of corresponding partial sums converges to 0 uniformly for $-1\le x\le 1$. …

36: 14.28 Sums

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

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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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§14.28(iii) Other Sums

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37: 27.14 Unrestricted Partitions

… ►A fundamental problem studies the number of ways $n$ can be written as a sum of positive integers $\le n$, that is, the number of solutions of … ►and $s(h,k)$ is a Dedekind sum given by ►

27.14.11 $$s(h,k)=\sum _{r=1}^{k-1}\frac{r}{k}\left(\frac{hr}{k}-\lfloor \frac{hr}{k}\rfloor -\frac{1}{2}\right).$$

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Defines:

$s(h,k)$: Dedekind sum (locally)

Symbols:

$\lfloor x\rfloor$: floor of $x$ and $k$: positive integer

A&S Ref:

24.2.1 I.C

Referenced by:

§23.18, §27.14(iv)

Permalink:

http://dlmf.nist.gov/27.14.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(iii), §27.14 and Ch.27

… ►Dedekind sums occur in the transformation theory of the Dedekind modular function $\eta \left(\tau \right)$, defined by …where $\epsilon =\exp \left(\pi \mathrm{i}(((a+d)/(12c))-s(d,c))\right)$ and $s(d,c)$ is given by (27.14.11). …

38: 31.14 General Fuchsian Equation

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31.14.1 $$\frac{d^{2}w}{{dz}^2}+\left(\sum _{j=1}^N\frac{{\gamma }_j}{z-a_j}\right)\frac{dw}{dz}+\left(\sum _{j=1}^N\frac{q_j}{z-a_j}\right)w=0,$$ ${\sum }_{j=1}^N{q}_j=0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(i)

Permalink:

http://dlmf.nist.gov/31.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14 and Ch.31

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$\alpha +\beta +1={\displaystyle \sum _{j=1}^N}{\gamma }_j,$

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$\alpha \beta ={\displaystyle \sum _{j=1}^N}{a}_j{q}_j.$

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31.14.4 $$\frac{d^{2}W}{{dz}^2}=\sum _{j=1}^N\left(\frac{{\tilde{\gamma }}_j}{{(z-a_j)}^2}+\frac{{\tilde{q}}_j}{z-a_j}\right)W,$$ ${\sum }_{j=1}^N{\tilde{q}}_j=0$,

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Defines:

$W(z)$: solution (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

►

${\tilde{q}}_j={\displaystyle \frac{1}{2}}{\displaystyle \sum _{\begin{array}{c}k=1\\ k\ne j\end{array}}^N}{\displaystyle \frac{{\gamma }_j{\gamma }_k}{a_j-a_k}}-q_j,$

…

39: 6.6 Power Series

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6.6.1 $$\mathrm{Ei}\left(x\right)=\gamma +\ln x+\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n!n},$$ $x>0$.

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $x$: real variable and $n$: nonnegative integer

A&S Ref:

5.1.10

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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6.6.2 $$E_1\left(z\right)=-\gamma -\ln z-\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^n}{n!n}.$$

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Symbols:

$\gamma$: Euler’s constant, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.1.11

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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6.6.3 $$E_1\left(z\right)=-\ln z+{\mathrm{e}}^{-z}\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}\psi \left(n+1\right),$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§2.5(iii), §6.6, §8.19(iv)

Permalink:

http://dlmf.nist.gov/6.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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6.6.4 $$\mathrm{Ein}\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}{z}^n}{n!n},$$

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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6.6.5 $$\mathrm{Si}\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^{2n+1}}{(2n+1)!(2n+1)},$$

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Symbols:

$!$: factorial (as in $n!$), $\mathrm{Si}\left(z\right)$: sine integral, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.2.14

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

…

40: 10.53 Power Series

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10.53.1 $${𝗃}_n\left(z\right)=z^n\sum _{k=0}^{\mathrm{\infty }}\frac{{(-\frac{1}{2}{z}^2)}^k}{k!(2n+2k+1)!!},$$

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Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.1.2

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

►

10.53.2 $${𝗒}_n\left(z\right)=-\frac{1}{z^{n+1}}\sum _{k=0}^n\frac{(2n-2k-1)!!{(\frac{1}{2}{z}^2)}^k}{k!}+\frac{{(-1)}^{n+1}}{z^{n+1}}\sum _{k=n+1}^{\mathrm{\infty }}\frac{{(-\frac{1}{2}{z}^2)}^k}{k!(2k-2n-1)!!}.$$

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Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.1.3

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

►

10.53.3 $${𝗂}_n^{(1)}\left(z\right)=z^n\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}{z}^2)}^k}{k!(2n+2k+1)!!},$$

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Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.2.5

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

►

10.53.4 $${𝗂}_n^{(2)}\left(z\right)=\frac{{(-1)}^n}{z^{n+1}}\sum _{k=0}^n\frac{(2n-2k-1)!!{(-\frac{1}{2}{z}^2)}^k}{k!}+\frac{1}{z^{n+1}}\sum _{k=n+1}^{\mathrm{\infty }}\frac{{(\frac{1}{2}{z}^2)}^k}{k!(2k-2n-1)!!}.$$

ⓘ

Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.2.6

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

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