elementary identities

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11: 18.36 Miscellaneous Polynomials

… ►The possibility of generalization to $\alpha =-k$, for $k\in \mathbb{N}$, is implicit in the identity Szegő (1975, page 102), … ►See Gómez-Ullate et al. (2009) for an elementary introduction. … ►

18.36.7 $$T_k(y)\equiv -x{y}^{\prime \prime }+\frac{x-k}{x+k}((x+k+1)y^{\prime }-y)=(n-1)y.$$

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Symbols:

$\equiv$: equals by definition, $y$: real variable, $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.36(vi)

Permalink:

http://dlmf.nist.gov/18.36.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

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12: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►

§1.3(i) Determinants: Elementary Properties

… ►If two rows (columns) of a determinant are identical, then the determinant is zero. …

13: 25.5 Integral Representations

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§25.5(i) In Terms of Elementary Functions

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25.5.6 $$\zeta \left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{{\mathrm{e}}^x}dx,$$ $\mathrm{\Re }s>-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Proof sketch:

Derivable from (25.5.1) by first replacing $x^{s-1}\mapsto x^{s-1}(1+{\mathrm{e}}^{-x}-{\mathrm{e}}^{-x})$, using the identity ${\mathrm{e}}^{-x}=(1-{\mathrm{e}}^{-x})/({\mathrm{e}}^x-1)$, and replacing ${({\mathrm{e}}^x-1)}^{-1}\mapsto ({({\mathrm{e}}^x-1)}^{-1}-1/x+1/2-1/2+1/x)$ in the resulting integrand with (5.2.1), (5.5.1).

Referenced by:

(25.5.7)

Permalink:

http://dlmf.nist.gov/25.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(i), §25.5 and Ch.25

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25.5.14 $$\omega (x)\equiv \sum _{n=1}^{\mathrm{\infty }}{\mathrm{e}}^{-n^2\pi x}=\frac{1}{2}\left({\theta }_3\left(0\right|\mathrm{i}x)-1\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer and $x$: real variable

Keywords:

definition

Source:

Titchmarsh (1986b, p. 22)

Permalink:

http://dlmf.nist.gov/25.5.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(ii), §25.5 and Ch.25

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14: 25.12 Polylogarithms

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25.12.1 $${\mathrm{Li}}_2\left(z\right)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n^2},$$ $|z|\le 1$.

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Defines:

${\mathrm{Li}}_2\left(z\right)$: dilogarithm

Symbols:

$\equiv$: equals by definition, $n$: nonnegative integer, $x$: real variable and $z$: complex variable

Keywords:

definition

Source:

Maximon (2003, (3.1), p. 2808)

A&S Ref:

27.7.2 (with $z=1-x$)

Referenced by:

§25.12(i), §25.12(i)

Permalink:

http://dlmf.nist.gov/25.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

… ►The cosine series in (25.12.7) has the elementary sum … ►

25.12.11 $${\mathrm{Li}}_s\left(z\right)\equiv \frac{z}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}}{{\mathrm{e}}^x-z}dx,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\int$: integral, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $x$: real variable, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Lewin (1981, (7.188), p. 236)

Referenced by:

(25.12.16), §25.12(ii)

Permalink:

http://dlmf.nist.gov/25.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

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15: 15.4 Special Cases

… ►

§15.4(i) Elementary Functions

… ►For an extensive list of elementary representations see Prudnikov et al. (1990, pp. 468–488). … ►

Chu–Vandermonde Identity

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