interrelation between bases

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1: 31.16 Mathematical Applications

… ►Expansions of Heun polynomial products in terms of Jacobi polynomial (§18.3) products are derived in Kalnins and Miller (1991a, b, 1993) from the viewpoint of interrelation between two bases in a Hilbert space: … ►

2: 6.5 Further Interrelations

§6.5 Further Interrelations

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6.5.7 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=E_1\left(\mp \mathrm{i}z\right){\mathrm{e}}^{\mp \mathrm{i}z}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.11, §6.5, §6.7(iii)

Permalink:

http://dlmf.nist.gov/6.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

3: 7.5 Interrelations

§7.5 Interrelations

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7.5.1 $$F\left(z\right)=\frac{1}{2}\mathrm{i}\sqrt{\pi }\left({\mathrm{e}}^{-z^2}-w\left(z\right)\right)=-\frac{1}{2}\mathrm{i}\sqrt{\pi }{\mathrm{e}}^{-z^2}\mathrm{erf}\left(\mathrm{i}z\right).$$

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Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.5 $${\mathrm{e}}^{-\frac{1}{2}\pi \mathrm{i}{z}^2}ℱ\left(z\right)=\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right).$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $ℱ\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.13(iv), §7.5, §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.10 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=\frac{1}{2}(1\pm \mathrm{i}){\mathrm{e}}^{{\zeta }^2}\mathrm{erfc}\zeta .$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

A&S Ref:

7.3.22 (in different form)

Referenced by:

§7.12(ii), §7.13(iv), §7.5, §7.5

Permalink:

http://dlmf.nist.gov/7.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.13 $$G\left(x\right)=\sqrt{\pi }F\left(x\right)-\frac{1}{2}{\mathrm{e}}^{-x^2}\mathrm{Ei}\left(x^2\right),$$ $x>0$.

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Symbols:

$F\left(z\right)$: Dawson’s integral, $G\left(z\right)$: Goodwin–Staton integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{Ei}\left(x\right)$: exponential integral and $x$: real variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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4: 18.21 Hahn Class: Interrelations

§18.21 Hahn Class: Interrelations

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Hahn $\to$ Jacobi

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Meixner $\to$ Laguerre

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Charlier $\to$ Hermite

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Meixner–Pollaczek $\to$ Laguerre

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5: 23.19 Interrelations

§23.19 Interrelations

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6: 13.2 Definitions and Basic Properties

… ► … ►

13.2.7 $$U(-m,b,z)={(-1)}^m{\left(b\right)}_{m}M(-m,b,z)={(-1)}^m\sum _{s=0}^m\left(\genfrac{}{}{0.0pt}{}{m}{s}\right){\left(b+s\right)}_{m-s}{(-z)}^s.$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $m$: integer, $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(i), Erratum (V1.0.10) for Equation (13.2.7)

Permalink:

http://dlmf.nist.gov/13.2.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

The equality $U(-m,b,z)={(-1)}^m{\left(b\right)}_{m}M(-m,b,z)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation $a=-n$ has been changed to $a=-m$.

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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13.2.8 $$U(a,a+n+1,z)=\frac{{(-1)}^n{\left(1-a-n\right)}_n}{z^{a+n}}M(-n,1-a-n,z)=z^{-a}\sum _{s=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{s}\right){\left(a\right)}_s{z}^{-s}.$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(i), Erratum (V1.0.10) for Equation (13.2.8)

Permalink:

http://dlmf.nist.gov/13.2.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

The equality $U(a,a+n+1,z)=\frac{{(-1)}^n{\left(1-a-n\right)}_n}{z^{a+n}}M(-n,1-a-n,z)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation.

Suggested 2014-02-10 by Adri Olde Daalhuis

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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13.2.10 $$U(-m,n+1,z)={(-1)}^m{\left(n+1\right)}_{m}M(-m,n+1,z)={(-1)}^m\sum _{s=0}^m\left(\genfrac{}{}{0.0pt}{}{m}{s}\right){\left(n+s+1\right)}_{m-s}{(-z)}^s.$$

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $m$: integer, $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(i), Erratum (V1.0.10) for Equation (13.2.10), Erratum (V1.0.12) for Equations (13.2.9), (13.2.10)

Permalink:

http://dlmf.nist.gov/13.2.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.12):

. The condition $a=-m,m=0,1,2,\mathrm{\dots }$ that appeared below this equation now appears ahead of the equation.

Suggested 2016-07-05 by Adri Olde Daalhuis

Addition (effective with 1.0.10):

The equality $U(-m,n+1,z)={(-1)}^m{\left(n+1\right)}_{m}M(-m,n+1,z)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation $a=-m,m=0,1,2,\mathrm{\dots }$ has been introduced.

Suggested 2015-02-10 by Adri Olde Daalhuis

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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Kummer’s Transformations

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7: 4.37 Inverse Hyperbolic Functions

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§4.37(vi) Interrelations

►Table 4.30.1 can also be used to find interrelations between inverse hyperbolic functions. …

8: 6.2 Definitions and Interrelations

§6.2 Definitions and Interrelations

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6.2.1 $$E_1\left(z\right)={\int }_z^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{t}dt,$$ $z\ne 0$,

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Defines:

$E_1\left(z\right)$: exponential integral

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $z$: complex variable

A&S Ref:

5.1.1

Referenced by:

§6.2(ii), §6.7(iii)

Permalink:

http://dlmf.nist.gov/6.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

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6.2.3 $$\mathrm{Ein}\left(z\right)={\int }_0^z\frac{1-{\mathrm{e}}^{-t}}{t}dt.$$

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Defines:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $z$: complex variable

Permalink:

http://dlmf.nist.gov/6.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

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6.2.5 $$\mathrm{Ei}\left(x\right)=-{⨍}_{-x}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{t}dt={⨍}_{-\mathrm{\infty }}^x\frac{{\mathrm{e}}^t}{t}dt,$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{Ei}\left(x\right)$: exponential integral, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

A&S Ref:

5.1.2

Permalink:

http://dlmf.nist.gov/6.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

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9: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

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§18.7(i) Linear Transformations

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Legendre, Ultraspherical, and Jacobi

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§18.7(ii) Quadratic Transformations

… ►Equations (18.7.13)–(18.7.20) are special cases of (18.2.22)–(18.2.23). …

10: 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. …