transformations of variables

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1: 16.6 Transformations of Variable

§16.6 Transformations of Variable

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Quadratic

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Cubic

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16.6.2 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,2b-a-1,2-2b+a}{b,a-b+\frac{3}{2}};\frac{z}{4})={(1-z)}^{-a}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{3}a+\frac{2}{3}}{b,a-b+\frac{3}{2}};\frac{-27z}{4{(1-z)}^3}).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.6

Permalink:

http://dlmf.nist.gov/16.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.6, §16.6 and Ch.16

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2: Vadim B. Kuznetsov

… ►Kuznetsov published papers on special functions and orthogonal polynomials, the quantum scattering method, integrable discrete many-body systems, separation of variables, Bäcklund transformation techniques, and integrability in classical and quantum mechanics. …

3: 16.16 Transformations of Variables

§16.16 Transformations of Variables

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§16.16(i) Reduction Formulas

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16.16.10 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)=\frac{\mathrm{\Gamma }\left({\gamma }^{\prime }\right)\mathrm{\Gamma }\left(\beta -\alpha \right)}{\mathrm{\Gamma }\left({\gamma }^{\prime }-\alpha \right)\mathrm{\Gamma }\left(\beta \right)}{(-y)}^{-\alpha }{F}_4(\alpha ,\alpha -{\gamma }^{\prime }+1;\gamma ,\alpha -\beta +1;\frac{x}{y},\frac{1}{y})+\frac{\mathrm{\Gamma }\left({\gamma }^{\prime }\right)\mathrm{\Gamma }\left(\alpha -\beta \right)}{\mathrm{\Gamma }\left({\gamma }^{\prime }-\beta \right)\mathrm{\Gamma }\left(\alpha \right)}{(-y)}^{-\beta }{F}_4(\beta ,\beta -{\gamma }^{\prime }+1;\gamma ,\beta -\alpha +1;\frac{x}{y},\frac{1}{y}).$$

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Symbols:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function and $\mathrm{\Gamma }\left(z\right)$: gamma function

Permalink:

http://dlmf.nist.gov/16.16.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(ii), §16.16 and Ch.16

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4: 15.19 Methods of Computation

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5: 15.8 Transformations of Variable

§15.8 Transformations of Variable

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§15.8(i) Linear Transformations

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§15.8(iii) Quadratic Transformations

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§15.8(v) Cubic Transformations

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6: 18.25 Wilson Class: Definitions

… ►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials $W_n(x;a,b,c,d)$, continuous dual Hahn polynomials $S_n(x;a,b,c)$, Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta )$, and dual Hahn polynomials $R_n(x;\gamma ,\delta ,N)$. ►

Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.

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OP	$p_n(x)$	$x=\lambda (y)$	Orthogonality range for $y$	Constraints
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7: 23.18 Modular Transformations

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23.18.3 $$\lambda \left(𝒜\tau \right)=\lambda \left(\tau \right),$$

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Symbols:

$\lambda \left(\tau \right)$: elliptic modular function, $\tau$: complex variable and $𝒜$: bilinear transformation

Permalink:

http://dlmf.nist.gov/23.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

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23.18.4 $$J\left(𝒜\tau \right)=J\left(\tau \right).$$

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Symbols:

$J\left(\tau \right)$: Klein’s complete invariant, $\tau$: complex variable and $𝒜$: bilinear transformation

Referenced by:

§23.18

Permalink:

http://dlmf.nist.gov/23.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

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23.18.5 $$\eta \left(𝒜\tau \right)=\epsilon (𝒜){\left(-\mathrm{i}(c\tau +d)\right)}^{1/2}\eta \left(\tau \right),$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\mathrm{i}$: imaginary unit, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer and $\epsilon (𝒜)$: function

Permalink:

http://dlmf.nist.gov/23.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

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8: 31.7 Relations to Other Functions

… ►Joyce (1994) gives a reduction in which the independent variable is transformed not polynomially or rationally, but algebraically. …

9: 23.15 Definitions

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23.15.3 $$𝒜\tau =\frac{a\tau +b}{c\tau +d},$$

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Symbols:

$\tau$: complex variable, $𝒜$: bilinear transformation, $a$: integer, $b$: integer, $c$: integer and $d$: integer

Referenced by:

§20.11(i), §23.18

Permalink:

http://dlmf.nist.gov/23.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(i), §23.15 and Ch.23

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23.15.5 $$f(𝒜\tau )=c_{𝒜}{(c\tau +d)}^{\mathrm{\ell }}f(\tau ),$$ $\mathrm{\Im }\tau >0$,

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Symbols:

$\mathrm{\Im }$: imaginary part, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer, $f(\tau )$: modular function, $c_{𝒜}$: constant and $\mathrm{\ell }$: level

Permalink:

http://dlmf.nist.gov/23.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(i), §23.15 and Ch.23

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10: 1.13 Differential Equations

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Transformation of the Point at Infinity

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Liouville Transformation

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