addition formulas

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11: 25.16 Mathematical Applications

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25.16.12 $$H(s,z)+H(z,s)=\zeta \left(s\right)\zeta \left(z\right)+\zeta \left(s+z\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $H(s,z)$: generalized Euler sums, $s$: complex variable and $z$: complex variable

Keywords:

additionformula, connection formula

Source:

Apostol and Vu (1984, (11), p. 91)

Permalink:

http://dlmf.nist.gov/25.16.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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12: 19.26 Addition Theorems

§19.26 Addition Theorems

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§19.26(i) General Formulas

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§19.26(iii) Duplication Formulas

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19.26.27 $$R_C(x^2,x^2-\theta )=2R_C(s^2,s^2-\theta ),$$ $s=x+\sqrt{x^2-\theta }$, $\theta \ne x^2$ or $s^2$.

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $x$: positive

Permalink:

http://dlmf.nist.gov/19.26.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(iii), §19.26 and Ch.19

13: 18.17 Integrals

… ►For addition formulas corresponding to (18.17.5) and (18.17.6) see (18.18.8) and (18.18.9), respectively. …

14: 30.10 Series and Integrals

… ►For product formulas and convolutions see Connett et al. (1993). For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …

15: 18.3 Definitions

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As given by a Rodrigues formula (18.5.5).

… ►For representations of the polynomials in Table 18.3.1 by Rodrigues formulas, see §18.5(ii). … ►In this chapter, formulas for the Chebyshev polynomials of the second, third, and fourth kinds will not be given as extensively as those of the first kind. However, most of these formulas can be obtained by specialization of formulas for Jacobi polynomials, via (18.7.4)–(18.7.6). … ►Formula (18.3.1) can be understood as a Gauss-Chebyshev quadrature, see (3.5.22), (3.5.23). …

16: 19.11 Addition Theorems

§19.11 Addition Theorems

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§19.11(i) General Formulas

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§19.11(iii) Duplication Formulas

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17: 3.3 Interpolation

… ►Newton’s formula has the advantage of allowing easy updating: incorporation of a new point $z_{n+1}$ requires only addition of the term with $\left[z_0,z_1,\mathrm{\dots },z_{n+1}\right]f$ to (3.3.38), plus the computation of this divided difference. …

18: 5.9 Integral Representations

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5.9.2_5 $$\frac{1}{\mathrm{\Gamma }\left(z\right)}=\frac{{\mathrm{e}}^z{z}^{1-z}}{2\pi }{\int }_{-\pi }^{\pi }{\mathrm{e}}^{-z\mathrm{\Phi }(t)}dt,$$ $\mathrm{\Re }z>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $\mathrm{\Re }$: real part, $z$: complex variable and $\mathrm{\Phi }(t)$: function

Source:

Temme (1996b, (3.38), p. 71)

Referenced by:

§5.9(i), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/5.9.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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Binet’s Formula

… ►Two alternative versions of Binet’s formula are ►

5.9.10_1 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)-\frac{z}{\pi }{\int }_0^{\mathrm{\infty }}\frac{\ln \left(1-{\mathrm{e}}^{-2\pi t}\right)}{t^2+z^2}dt,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Proof sketch:

Can be obtained from (5.9.10) via integration by parts.

Referenced by:

§5.9(i), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/5.9.E10_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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5.9.10_2 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}\left(\frac{1}{{\mathrm{e}}^t-1}-\frac{1}{t}+\frac{1}{2}\right)\frac{dt}{t},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Source:

Whittaker and Watson (1927, §12.31)

Referenced by:

§5.9(i), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/5.9.E10_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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19: 18.18 Sums

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§18.18(ii) Addition Theorems

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Ultraspherical

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Legendre

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§18.18(v) Linearization Formulas

… ►Formula (18.18.27) is known as the Hille–Hardy formula. …

20: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

… ►Related formulas are (17.7.3), (17.8.8) and ►

17.6.4_5 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{b^2,b^2/c}{c{q}^2};q^2,c{q}^3/b^2)=\frac{1}{2b}\frac{{(b^2,q;q^2)}_{\mathrm{\infty }}}{{(c{q}^2,cq/b^2;q^2)}_{\mathrm{\infty }}}\left(\frac{{(cq/b;q)}_{\mathrm{\infty }}}{{(b;q)}_{\mathrm{\infty }}}-\frac{{(-cq/b;q)}_{\mathrm{\infty }}}{{(-b;q)}_{\mathrm{\infty }}}\right),$$ .

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Source:

Verma and Jain (1983, (3.6))

Referenced by:

§17.6(i), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/17.6.E4_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

Suggested 2019-10-19 by Slobodan Damjanovic

See also:

Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

►For similar formulas see Verma and Jain (1983). …