Hypergeometric function

The hypergeometric function ( Gauss function ) is defined inside the circle $|z|<1$ ${\ displaystyle | z | <1}$ $| z | <1$ as the sum of the hypergeometric series

F(a,b;c;z)=1+\sum _{k=1}^{\infty }\left[\prod _{l=0}^{k-1}{(a+l)(b+l) \over (1+l)(c+l)}\right]z^{k}=1+{\frac {ab}{c}}{\frac {z}{1!}}+{\frac {a(a+1)b(b+1)}{c(c+1)}}{\frac {z^{2}}{2!}}+\dots ,

{\ displaystyle F (a, b; c; z) = 1 + \ sum _ {k = 1} ^ {\ infty} \ left [\ prod _ {l = 0} ^ {k-1} {(a + l) (b + l) \ over (1 + l) (c + l)} \ right] z ^ {k} = 1 + {\ frac {ab} {c}} {\ frac {z} {1! }} + {\ frac {a (a + 1) b (b + 1)} {c (c + 1)}} {\ frac {z ^ {2}} {2!}} + \ dots,}

F (a, b; c; z) = 1 + \ sum _ {{k = 1}} ^ {\ infty} \ left [\ prod _ {{l = 0}} ^ {{k-1}} { (a + l) (b + l) \ over (1 + l) (c + l)} \ right] z ^ {k} = 1 + {\ frac {ab} {c}} {\ frac {z} {1!}} + {\ Frac {a (a + 1) b (b + 1)} {c (c + 1)}} {\ frac {z ^ {2}} {2!}} + \ Dots ,

and when $|z|>1$ ${\ displaystyle | z |> 1}$ $| z |> 1$ - as its analytical continuation . It is a solution to a linear ordinary differential equation (ODE) of the second order, called the hypergeometric equation.

Content

1 History
2 Hypergeometric equation
3 particular values for $z=1/2$ ${\ displaystyle z = 1/2}$ ${\ displaystyle z = 1/2}$
4 Recording other functions through hypergeometric
- 4.1 Examples
5 Identities
6 notes
7 Literature

History

The term "hypergeometric series" was first used by John Wallis in 1655 in the book Arithmetica Infinitorum . This term refers to a series whose general formula of members has the form ^[1]

{\frac {1\cdot 3\cdot 5\cdot \ldots \cdot (2n+1)}{2\cdot 4\cdot \ldots \cdot 2n}}.

{\ displaystyle {\ frac {1 \ cdot 3 \ cdot 5 \ cdot \ ldots \ cdot (2n + 1)} {2 \ cdot 4 \ cdot \ ldots \ cdot 2n}}.}

{\frac {1\cdot 3\cdot 5\cdot \ldots \cdot (2n+1)}{2\cdot 4\cdot \ldots \cdot 2n}}.

Hypergeometric series were studied by Leonard Euler , and in more detail by Gauss ^[2] . In the 19th century, the study was continued by Ernst Kummer, and Bernhard Riemann defined a hypergeometric function through an equation that it satisfies.

Hypergeometric equation

Consider the Euler differential equation $z(1-z){\frac {d^{2}u}{dz^{2}}}+[c-(a+b+1)z]{\frac {du}{dz}}-abu=0,$ ${\ displaystyle z (1-z) {\ frac {d ^ {2} u} {dz ^ {2}}} + [c- (a + b + 1) z] {\ frac {du} {dz} } -abu = 0,}$ $z(1-z){\frac {d^{2}u}{dz^{2}}}+[c-(a+b+1)z]{\frac {du}{dz}}-abu=0,$ where the parameters a , b, and c can be arbitrary complex numbers. Its generalization to arbitrary regular singular points is given by the Riemann differential equation . The Euler equation has three singular points : 0, 1 and $\infty$ ${\ displaystyle \ infty}$ $\infty$ .

When parameter $c$ ${\ displaystyle c}$ $c$ non-zero and negative integers $(c\neq 0,-1,-2,\ldots )$ ${\ displaystyle (c \ neq 0, -1, -2, \ ldots)}$ $(c\neq 0,-1,-2,\ldots )$ a regular solution to zero of the Euler equation can be written in a series called hypergeometric:

_{2}F_{1}(a,b;c;z)\equiv F(a,b;c;z)=1+{\frac {ab}{c}}{\frac {z}{1!}}+{\frac {a(a+1)b(b+1)}{c(c+1)}}{\frac {z^{2}}{2!}}+\dots .

{\ displaystyle _ {2} F_ {1} (a, b; c; z) \ equiv F (a, b; c; z) = 1 + {\ frac {ab} {c}} {\ frac {z } {1!}} + {\ Frac {a (a + 1) b (b + 1)} {c (c + 1)}} {\ frac {z ^ {2}} {2!}} + \ dots.}

_{2}F_{1}(a,b;c;z)\equiv F(a,b;c;z)=1+{\frac {ab}{c}}{\frac {z}{1!}}+{\frac {a(a+1)b(b+1)}{c(c+1)}}{\frac {z^{2}}{2!}}+\dots .

This function is called hypergeometric. Often the designation is used ( Pohhammer symbol )

(p)_{n}={\frac {\Gamma (p+n)}{\Gamma (p)}},

{\ displaystyle (p) _ {n} = {\ frac {\ Gamma (p + n)} {\ Gamma (p)}},}

Where $\Gamma$ ${\ displaystyle \ Gamma}$ - gamma function . Then the hypergeometric function can be represented as

F(a,b;c;z)=\sum _{n=0}^{\infty }{\frac {(a)_{n}(b)_{n}z^{n}}{(c)_{n}n!}}.

{\ displaystyle F (a, b; c; z) = \ sum _ {n = 0} ^ {\ infty} {\ frac {(a) _ {n} (b) _ {n} z ^ {n} } {(c) _ {n} n!}}.}

Designation $_{2}F_{1}(a,b;c;z)$ ${\ displaystyle _ {2} F_ {1} (a, b; c; z)}$ indicate that there are two parameters, a and b, "going to the numerator", and one, c, "going to the denominator." On the border $|z|=1$ ${\ displaystyle | z | = 1}$ the series through which the hypergeometric function is determined absolutely converges if the real part of the sum $a+b-c<0$ ${\ displaystyle a + bc <0}$ conditionally converges at $z\neq 1$ ${\ displaystyle z \ neq 1}$ , $0\leq a+b-c<1$ ${\ displaystyle 0 \ leq a + bc <1}$ and diverges if $a+b-c\geq 1$ ${\ displaystyle a + bc \ geq 1}$ . The second linearly independent solution of the Euler differential equation has the form

\ z^{1-c}F(b-c+1,a-c+1;2-c;z)

{\ displaystyle \ z ^ {1-c} F (b-c + 1, a-c + 1; 2-c; z)}

It has a singular point for $z=0$ ${\ displaystyle z = 0}$ and fair for all non-positive $c$ ${\ displaystyle c}$ $(c=0,-1,-2,\ldots )$ ${\ displaystyle (c = 0, -1, -2, \ ldots)}$ . ^[3]

Integral representation for hypergeometric function for ${\text{Re}}(c)>{\text{Re}}(b)>0$ ${\ displaystyle {\ text {Re}} (c)> {\ text {Re}} (b)> 0}$ (Euler's formula) can be written as follows:

F(a,b;c;z)={\Gamma (c) \over \Gamma (b)\Gamma (c-b)}\int \limits _{0}^{1}t^{b-1}(1-t)^{c-b-1}(1-tz)^{-a}\,dt,

{\ displaystyle F (a, b; c; z) = {\ Gamma (c) \ over \ Gamma (b) \ Gamma (cb)} \ int \ limits _ {0} ^ {1} t ^ {b- 1} (1-t) ^ {cb-1} (1-tz) ^ {- a} \, dt,}

Where $\Gamma (x)$ ${\ displaystyle \ Gamma (x)}$ - gamma function of Euler. This expression is a unique analytic function on the complex $z$ ${\ displaystyle z}$ planes with a cut along the real axis from $one$ ${\ displaystyle 1}$ before $\infty$ ${\ displaystyle \ infty}$ and provides an analytic continuation to the entire complex plane for the hypergeometric series that converges only when $\left|z\right|<1$ ${\ displaystyle \ left | z \ right | <1}$ .

Particular values for $z=1/2$ ${\ displaystyle z = 1/2}$ ${\ displaystyle z = 1/2}$

The second Gauss summation theorem is expressed by the formula:

_{2}F_{1}\left(a,b;{\tfrac {1}{2}}\left(1+a+b\right);{\tfrac {1}{2}}\right)={\frac {\Gamma ({\tfrac {1}{2}})\Gamma ({\tfrac {1}{2}}\left(1+a+b\right))}{\Gamma ({\tfrac {1}{2}}\left(1+a)\right)\Gamma ({\tfrac {1}{2}}\left(1+b\right))}}.

{\ displaystyle _ {2} F_ {1} \ left (a, b; {\ tfrac {1} {2}} \ left (1 + a + b \ right); {\ tfrac {1} {2}} \ right) = {\ frac {\ Gamma ({\ tfrac {1} {2}}) \ Gamma ({\ tfrac {1} {2}} \ left (1 + a + b \ right))} {\ Gamma ({\ tfrac {1} {2}} \ left (1 + a) \ right) \ Gamma ({\ tfrac {1} {2}} \ left (1 + b \ right))}}.}

Bailey's theorem is expressed by the formula:

_{2}F_{1}\left(a,1-a;c;{\tfrac {1}{2}}\right)={\frac {\Gamma ({\tfrac {1}{2}}c)\Gamma ({\tfrac {1}{2}}\left(1+c\right))}{\Gamma ({\tfrac {1}{2}}\left(c+a\right))\Gamma ({\tfrac {1}{2}}\left(1+c-a\right))}}.

{\ displaystyle _ {2} F_ {1} \ left (a, 1-a; c; {\ tfrac {1} {2}} \ right) = {\ frac {\ Gamma ({\ tfrac {1} { 2}} c) \ Gamma ({\ tfrac {1} {2}} \ left (1 + c \ right))} {\ Gamma ({\ tfrac {1} {2}} \ left (c + a \ right)) \ Gamma ({\ tfrac {1} {2}} \ left (1 + ca \ right))}}.}

Recording other functions through hypergeometric

An important property of a hypergeometric function is that many special and elementary functions can be obtained from it for certain parameter values and the transformation of an independent argument.

Examples

$\left(1+x\right)^{n}=F(-n,b;b;-x)$ ${\ displaystyle \ left (1 + x \ right) ^ {n} = F (-n, b; b; -x)}$
$x^{n}=F\left(-n,b;b;1-x\right)$ ${\ displaystyle x ^ {n} = F \ left (-n, b; b; 1-x \ right)}$
${1 \over x}\ln(1+x)=F(1,1;2;-x)$ ${\ displaystyle {1 \ over x} \ ln (1 + x) = F (1,1; 2; -x)}$

{1 \over x}\arcsin(x)=F\left({\frac {1}{2}},{\frac {1}{2}};{\frac {3}{2}};x^{2}\right)

{\ displaystyle {1 \ over x} \ arcsin (x) = F \ left ({\ frac {1} {2}}, {\ frac {1} {2}}; {\ frac {3} {2} }; x ^ {2} \ right)}

$e^{x}=\lim _{n\to \infty }F\left(1,n;1;{x \over n}\right)$ ${\ displaystyle e ^ {x} = \ lim _ {n \ to \ infty} F \ left (1, n; 1; {x \ over n} \ right)}$
$\cos x=\lim _{a,\;b\to \infty }F\left(a,b;{\frac {1}{2}};-{\frac {x^{2}}{4ab}}\right)$ ${\ displaystyle \ cos x = \ lim _ {a, \; b \ to \ infty} F \ left (a, b; {\ frac {1} {2}}; - {\ frac {x ^ {2} } {4ab}} \ right)}$
$\cosh x=\lim _{a,\;b\to \infty }F\left(a,b;{\frac {1}{2}};{x^{2} \over 4ab}\right)$ ${\ displaystyle \ cosh x = \ lim _ {a, \; b \ to \ infty} F \ left (a, b; {\ frac {1} {2}}; {x ^ {2} \ over 4ab} \ right)}$
Complete elliptic integral of the first kind:
$K(k)=\int \limits _{0}^{\pi /2}\!{\frac {d\varphi }{\sqrt {1-k^{2}\sin ^{2}\varphi }}}={\frac {\pi }{2}}F\left({\frac {1}{2}},{\frac {1}{2}};1;k^{2}\right)$ ${\ displaystyle K (k) = \ int \ limits _ {0} ^ {\ pi / 2} \! {\ frac {d \ varphi} {\ sqrt {1-k ^ {2} \ sin ^ {2} \ varphi}}} = {\ frac {\ pi} {2}} F \ left ({\ frac {1} {2}}, {\ frac {1} {2}}; 1; k ^ {2} \ right)}$
The complete elliptic integral of the second kind:
$E(k)=\int \limits _{0}^{\pi /2}\!{\sqrt {1-k^{2}\sin ^{2}\varphi }}\,d\varphi ={\frac {\pi }{2}}F\left(-{\frac {1}{2}},{\frac {1}{2}};1;k^{2}\right)$ ${\ displaystyle E (k) = \ int \ limits _ {0} ^ {\ pi / 2} \! {\ sqrt {1-k ^ {2} \ sin ^ {2} \ varphi}} \, d \ varphi = {\ frac {\ pi} {2}} F \ left (- {\ frac {1} {2}}, {\ frac {1} {2}}; 1; k ^ {2} \ right) }$
Legendre Polynomial :
$P_{n}(x)=F(n+1,-n;1;{\frac {1-x}{2}})$ ${\ displaystyle P_ {n} (x) = F (n + 1, -n; 1; {\ frac {1-x} {2}})}$
The attached Legendre function :
$P_{n,\;m}(x)=(1-x^{2})^{\frac {m}{2}}{\Gamma (n+m+1) \over 2^{m}\Gamma (n-m+1)\Gamma (m+1)}F\left(n+m+1,m-n;m+1;{\frac {1-x}{2}}\right)$ ${\ displaystyle P_ {n, \; m} (x) = (1-x ^ {2}) ^ {\ frac {m} {2}} {\ Gamma (n + m + 1) \ over 2 ^ { m} \ Gamma (n-m + 1) \ Gamma (m + 1)} F \ left (n + m + 1, mn; m + 1; {\ frac {1-x} {2}} \ right) }$
Bessel Functions :
$J_{\nu }(x)=\lim _{a,\;b\to \infty }\left[{\frac {\left({\dfrac {x}{2}}\right)^{\nu }}{\Gamma (\nu +1)}}F\left(a,b;\nu +1;-{\frac {x^{2}}{4ab}}\right)\right]$ ${\ displaystyle J _ {\ nu} (x) = \ lim _ {a, \; b \ to \ infty} \ left [{\ frac {\ left ({\ dfrac {x} {2}} \ right) ^ {\ nu}} {\ Gamma (\ nu +1)}} F \ left (a, b; \ nu +1; - {\ frac {x ^ {2}} {4ab}} \ right) \ right] }$
Kummer (Pohhammer) function , or degenerate hypergeometric function
$M(a,c,z)={}_{1}F_{1}(a,c,z)=\lim _{b\to \infty }F(a,b;c;z/b)$ ${\ displaystyle M (a, c, z) = {} _ {1} F_ {1} (a, c, z) = \ lim _ {b \ to \ infty} F (a, b; c; z / b)}$
is a solution to a degenerate hypergeometric equation
$z{\frac {d^{2}w}{dz^{2}}}+(c-z){\frac {dw}{dz}}-aw=0.$ ${\ displaystyle z {\ frac {d ^ {2} w} {dz ^ {2}}} + (cz) {\ frac {dw} {dz}} - aw = 0.}$
A degenerate hypergeometric function with an integer nonpositive first argument is a generalized Laguerre polynomial :
$L_{n}^{\lambda }(x)={}_{1}F_{1}(-n,\lambda ,x).$ ${\ displaystyle L_ {n} ^ {\ lambda} (x) = {} _ {1} F_ {1} (- n, \ lambda, x).}$

Identities

$27\,(z-1)^{2}\cdot {_{2}F_{1}}\left({\tfrac {1}{4}},{\tfrac {3}{4}};{\tfrac {2}{3}};z\right)^{8}+18\,(z-1)\cdot {_{2}F_{1}}\left({\tfrac {1}{4}},{\tfrac {3}{4}};{\tfrac {2}{3}};z\right)^{4}-8\cdot {_{2}F_{1}}\left({\tfrac {1}{4}},{\tfrac {3}{4}};{\tfrac {2}{3}};z\right)^{2}=1$ ${\ displaystyle 27 \, (z-1) ^ {2} \ cdot {_ {2} F_ {1}} \ left ({\ tfrac {1} {4}}, {\ tfrac {3} {4} }; {\ tfrac {2} {3}}; z \ right) ^ {8} +18 \, (z-1) \ cdot {_ {2} F_ {1}} \ left ({\ tfrac {1 } {4}}, {\ tfrac {3} {4}}; {\ tfrac {2} {3}}; z \ right) ^ {4} -8 \ cdot {_ {2} F_ {1}} \ left ({\ tfrac {1} {4}}, {\ tfrac {3} {4}}; {\ tfrac {2} {3}}; z \ right) ^ {2} = 1}$
And a wonderful special case of the previous expression:
$_{2}F_{1}\left({\frac {1}{4}},{\frac {3}{4}};\,{\frac {2}{3}};\,{\frac {1}{3}}\right)={\frac {1}{\sqrt {{\sqrt {{\frac {4}{\sqrt {2-{\sqrt[{3}]{4}}}}}+{\sqrt[{3}]{4}}+4}}-{\sqrt {2-{\sqrt[{3}]{4}}}}-2}}}$ ${\ displaystyle _ {2} F_ {1} \ left ({\ frac {1} {4}}, {\ frac {3} {4}}; \, {\ frac {2} {3}}; \ , {\ frac {1} {3}} \ right) = {\ frac {1} {\ sqrt {{\ sqrt {{\ frac {4} {\ sqrt {2 - {\ sqrt [{3}] { 4}}}}} + {\ sqrt [{3}] {4}} + 4}} - {\ sqrt {2 - {\ sqrt [{3}] {4}}}} - 2}}}}$

Notes

↑ Scott, 1981 , p. 16.
↑ Vinogradov, 1977 , p. 1004.
↑ Bateman, Erdeyi, T. 1, 1973 , p. 69-70.

Literature

Mathematical Encyclopedia / Ed. I.M. Vinogradova. - M. , 1977 .-- T. 1.
Beitman G., Erdeyi A. Higher Transcendental Functions = Higher Transcendental Functions / Per. N. Ya. Vilenkina. - Ed. 2nd. - M .: Nauka, 1973. - T. 1. - 296 p. - 14,000 copies.
Kuznetsov D. S .: Special functions - M.: "Higher school", 1962
Landau, L.D. , Lifshits, E.M. Quantum mechanics (nonrelativistic theory). - 4th edition. - M .: Nauka , 1989 .-- 768 p. - (“ Theoretical Physics ”, Volume III). - ISBN 5-02-014421-5 . - mathematical additions
Kazuhiko Aomoto, Michitake Kita. Theory of Hypergeometric Functions / Transl. by Kenji Iohara. - Springer, 2011 .-- Vol. 305. - 317 p. - (Springer Monographs in Mathematics Series). - ISBN 9784431539124 .
Scott JF The mathematical work of John Wallis, DD, FRS, (1616-1703). - American Mathematical Soc., 1981. - 240 p. - (Chelsea Publishing Series). - ISBN 9780828403146 .

[_08e96690a951a86e-1] Scott, 1981 , p. 16.

[_9dfcac167eeb1371-2] Vinogradov, 1977 , p. 1004.

[_a7755a858d9539f4-3] Bateman, Erdeyi, T. 1, 1973 , p. 69-70.