Generalized hypergeometric function

The following basic identity is very useful as it relates the higher-order hypergeometric functions in terms of integrals over the lower order ones[2]

${\displaystyle {}_{A+1}F_{B+1}\left[{\begin{array}{c}a_{1},\ldots ,a_{A},c\\b_{1},\ldots ,b_{B},d\end{array}};z\right]={\frac {\Gamma (d)}{\Gamma (c)\Gamma (d-c)}}\int _{0}^{1}t^{c-1}(1-t)_{}^{d-c-1}\ {}_{A}F_{B}\left[{\begin{array}{c}a_{1},\ldots ,a_{A}\\b_{1},\ldots ,b_{B}\end{array}};tz\right]dt}$

The generalized hypergeometric function satisfies

${\displaystyle {\begin{aligned}\left(z{\frac {\rm {d}}{{\rm {d}}z}}+a_{j}\right){}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{j},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{array}};z\right]&=a_{j}\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{j}+1,\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{array}};z\right]\\\left(z{\frac {\rm {d}}{{\rm {d}}z}}+b_{k}-1\right){}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{k},\dots ,b_{q}\end{array}};z\right]&=(b_{k}-1)\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{k}-1,\dots ,b_{q}\end{array}};z\right]\\{\frac {\rm {d}}{{\rm {d}}z}}\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{array}};z\right]&={\frac {\prod _{i=1}^{p}a_{i}}{\prod _{j=1}^{q}b_{j}}}\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1}+1,\dots ,a_{p}+1\\b_{1}+1,\dots ,b_{q}+1\end{array}};z\right]\end{aligned}}}$

Combining these gives a differential equation satisfied by w = _pF_q:

${\displaystyle z\prod _{n=1}^{p}\left(z{\frac {\rm {d}}{{\rm {d}}z}}+a_{n}\right)w=z{\frac {\rm {d}}{{\rm {d}}z}}\prod _{n=1}^{q}\left(z{\frac {\rm {d}}{{\rm {d}}z}}+b_{n}-1\right)w}$.

Saalschütz's theorem[3] (Saalschütz 1890) is

${\displaystyle {}_{3}F_{2}(a,b,-n;c,1+a+b-c-n;1)={\frac {(c-a)_{n}(c-b)_{n}}{(c)_{n}(c-a-b)_{n}}}.}$

For extension of this theorem, see a research paper by Rakha & Rathie.

Dixon's identity,[4] first proved by Dixon (1902), gives the sum of a well-poised ₃F₂ at 1:

${\displaystyle {}_{3}F_{2}(a,b,c;1+a-b,1+a-c;1)={\frac {\Gamma (1+{\frac {a}{2}})\Gamma (1+{\frac {a}{2}}-b-c)\Gamma (1+a-b)\Gamma (1+a-c)}{\Gamma (1+a)\Gamma (1+a-b-c)\Gamma (1+{\frac {a}{2}}-b)\Gamma (1+{\frac {a}{2}}-c)}}.}$

For generalization of Dixon's identity, see a paper by Lavoie, et al.

Dougall's formula (Dougall 1907) gives the sum of a very well-poised series that is terminating and 2-balanced.

${\displaystyle {\begin{aligned}{}_{7}F_{6}&\left({\begin{matrix}a&1+{\frac {a}{2}}&b&c&d&e&-m\\&{\frac {a}{2}}&1+a-b&1+a-c&1+a-d&1+a-e&1+a+m\\\end{matrix}};1\right)=\\&={\frac {(1+a)_{m}(1+a-b-c)_{m}(1+a-c-d)_{m}(1+a-b-d)_{m}}{(1+a-b)_{m}(1+a-c)_{m}(1+a-d)_{m}(1+a-b-c-d)_{m}}}.\end{aligned}}}$

Terminating means that m is a non-negative integer and 2-balanced means that

${\displaystyle 1+2a=b+c+d+e-m.}$

Many of the other formulas for special values of hypergeometric functions can be derived from this as special or limiting cases.

Identity 1.

${\displaystyle e^{-x}\;{}_{2}F_{2}(a,1+d;c,d;x)={}_{2}F_{2}(c-a-1,f+1;c,f;-x)}$

where

${\displaystyle f={\frac {d(a-c+1)}{a-d}}}$;

Identity 2.

${\displaystyle e^{-{\frac {x}{2}}}\,{}_{2}F_{2}\left(a,1+b;2a+1,b;x\right)={}_{0}F_{1}\left(;a+{\tfrac {1}{2}};{\tfrac {x^{2}}{16}}\right)-{\frac {x\left(1-{\tfrac {2a}{b}}\right)}{2(2a+1)}}\;{}_{0}F_{1}\left(;a+{\tfrac {3}{2}};{\tfrac {x^{2}}{16}}\right),}$

which links Bessel functions to ₂F₂; this reduces to Kummer's second formula for b = 2a:

Identity 3.

${\displaystyle e^{-{\frac {x}{2}}}\,{}_{1}F_{1}(a,2a,x)={}_{0}F_{1}\left(;a+{\tfrac {1}{2}};{\tfrac {x^{2}}{16}}\right)}$.

Identity 4.

${\displaystyle {\begin{aligned}{}_{2}F_{2}(a,b;c,d;x)=&\sum _{i=0}{\frac {{b-d \choose i}{a+i-1 \choose i}}{{c+i-1 \choose i}{d+i-1 \choose i}}}\;{}_{1}F_{1}(a+i;c+i;x){\frac {x^{i}}{i!}}\\=&e^{x}\sum _{i=0}{\frac {{b-d \choose i}{a+i-1 \choose i}}{{c+i-1 \choose i}{d+i-1 \choose i}}}\;{}_{1}F_{1}(c-a;c+i;-x){\frac {x^{i}}{i!}},\end{aligned}}}$

which is a finite sum if b-d is a non-negative integer.

Kummer's relation is

${\displaystyle {}_{2}F_{1}\left(2a,2b;a+b+{\tfrac {1}{2}};x\right)={}_{2}F_{1}\left(a,b;a+b+{\tfrac {1}{2}};4x(1-x)\right).}$

Clausen's formula

${\displaystyle {}_{3}F_{2}(2c-2s-1,2s,c-{\tfrac {1}{2}};2c-1,c;x)=\,{}_{2}F_{1}(c-s-{\tfrac {1}{2}},s;c;x)^{2}}$

was used by de Branges to prove the Bieberbach conjecture.

As noted earlier, ${\displaystyle {}_{0}F_{0}(;;z)=e^{z}}$. The differential equation for this function is ${\displaystyle {\frac {d}{dz}}w=w}$, which has solutions ${\displaystyle w=ke^{z}}$ where k is a constant.

An important case is:

${\displaystyle {}_{1}F_{0}(a;;z)=(1-z)^{-a}.}$

The differential equation for this function is

${\displaystyle {\frac {d}{dz}}w=\left(z{\frac {d}{dz}}+a\right)w,}$

or

${\displaystyle (1-z){\frac {dw}{dz}}=aw,}$

which has solutions

${\displaystyle w=k(1-z)^{-a}}$

where k is a constant.

${\displaystyle {}_{1}F_{0}(1;;z)=\sum _{n\geqslant 0}z^{n}=(1-z)^{-1}}$ is the geometric series with ratio z and coefficient 1.

${\displaystyle z~{}_{1}F_{0}(2;;z)=\sum _{n\geqslant 0}nz^{n}=z(1-z)^{-2}}$ is also useful.

A special case is:

${\displaystyle {}_{0}F_{1}\left(;{\frac {1}{2}};-{\frac {z^{2}}{4}}\right)=\cos z}$

We can get this result, using the formula with rising factorials, as follows:

${\displaystyle {\begin{aligned}{}_{0}F_{1}\left(;{\tfrac {1}{2}};-{\tfrac {z^{2}}{4}}\right)&=\sum _{k=0}^{\infty }{\frac {1}{({\tfrac {1}{2}})_{k}}}\,{\frac {(-{\frac {z^{2}}{4}})^{k}}{k!}}=\sum _{k=0}^{\infty }{\frac {1}{\prod _{j=1}^{k}({\tfrac {1}{2}}+j-1)}}\,{\frac {(-{\frac {z^{2}}{4}})^{k}}{k!}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{k!4^{k}\prod _{j=1}^{k}(j-{\tfrac {1}{2}})}}\\&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{k!2^{k}2^{k}\prod _{j=1}^{k}({\tfrac {2j-1}{2}})}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{k!2^{k}2^{k}{\tfrac {\prod _{j=1}^{k}(2j-1)}{2^{k}}}}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{k!2^{k}\prod _{j=1}^{k}(2j-1)}}\\&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{\prod _{j=1}^{k}(2j)\prod _{j=1}^{k}(2j-1)}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k)!}}=\cos z\end{aligned}}}$

The functions of the form ${\displaystyle {}_{0}F_{1}(;a;z)}$ are called confluent hypergeometric limit functions and are closely related to Bessel functions.

The relationship is:

${\displaystyle J_{\alpha }(x)={\frac {({\tfrac {x}{2}})^{\alpha }}{\Gamma (\alpha +1)}}{}_{0}F_{1}\left(;\alpha +1;-{\tfrac {1}{4}}x^{2}\right).}$

${\displaystyle I_{\alpha }(x)={\frac {({\tfrac {x}{2}})^{\alpha }}{\Gamma (\alpha +1)}}{}_{0}F_{1}\left(;\alpha +1;{\tfrac {1}{4}}x^{2}\right).}$

The differential equation for this function is

${\displaystyle w=\left(z{\frac {d}{dz}}+a\right){\frac {dw}{dz}}}$

or

${\displaystyle z{\frac {d^{2}w}{dz^{2}}}+a{\frac {dw}{dz}}-w=0.}$

When a is not a positive integer, the substitution

${\displaystyle w=z^{1-a}u,}$

gives a linearly independent solution

${\displaystyle z^{1-a}\;{}_{0}F_{1}(;2-a;z),}$

so the general solution is

${\displaystyle k\;{}_{0}F_{1}(;a;z)+lz^{1-a}\;{}_{0}F_{1}(;2-a;z)}$

where k, l are constants. (If a is a positive integer, the independent solution is given by the appropriate Bessel function of the second kind.)

The functions of the form ${\displaystyle {}_{1}F_{1}(a;b;z)}$ are called confluent hypergeometric functions of the first kind, also written ${\displaystyle M(a;b;z)}$. The incomplete gamma function ${\displaystyle \gamma (a,z)}$ is a special case.

The differential equation for this function is

${\displaystyle \left(z{\frac {d}{dz}}+a\right)w=\left(z{\frac {d}{dz}}+b\right){\frac {dw}{dz}}}$

or

${\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0.}$

When b is not a positive integer, the substitution

${\displaystyle w=z^{1-b}u,}$

gives a linearly independent solution

${\displaystyle z^{1-b}\;{}_{1}F_{1}(1+a-b;2-b;z),}$

so the general solution is

${\displaystyle k\;{}_{1}F_{1}(a;b;z)+lz^{1-b}\;{}_{1}F_{1}(1+a-b;2-b;z)}$

where k, l are constants.

When a is a non-positive integer, −n, ${\displaystyle {}_{1}F_{1}(-n;b;z)}$ is a polynomial. Up to constant factors, these are the Laguerre polynomials. This implies Hermite polynomials can be expressed in terms of ₁F₁ as well.

This occurs in connection with the exponential integral function Ei(z).

Historically, the most important are the functions of the form ${\displaystyle {}_{2}F_{1}(a,b;c;z)}$. These are sometimes called Gauss's hypergeometric functions, classical standard hypergeometric or often simply hypergeometric functions. The term Generalized hypergeometric function is used for the functions _pF_q if there is risk of confusion. This function was first studied in detail by Carl Friedrich Gauss, who explored the conditions for its convergence.

The differential equation for this function is

${\displaystyle \left(z{\frac {d}{dz}}+a\right)\left(z{\frac {d}{dz}}+b\right)w=\left(z{\frac {d}{dz}}+c\right){\frac {dw}{dz}}}$

or

${\displaystyle z(1-z){\frac {d^{2}w}{dz^{2}}}+\left[c-(a+b+1)z\right]{\frac {dw}{dz}}-ab\,w=0.}$

It is known as the hypergeometric differential equation. When c is not a positive integer, the substitution

${\displaystyle w=z^{1-c}u}$

gives a linearly independent solution

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

so the general solution for |z| < 1 is

${\displaystyle k\;{}_{2}F_{1}(a,b;c;z)+lz^{1-c}\;{}_{2}F_{1}(1+a-c,1+b-c;2-c;z)}$

where k, l are constants. Different solutions can be derived for other values of z. In fact there are 24 solutions, known as the Kummer solutions, derivable using various identities, valid in different regions of the complex plane.

When a is a non-positive integer, −n,

${\displaystyle {}_{2}F_{1}(-n,b;c;z)}$

is a polynomial. Up to constant factors and scaling, these are the Jacobi polynomials. Several other classes of orthogonal polynomials, up to constant factors, are special cases of Jacobi polynomials, so these can be expressed using ₂F₁ as well. This includes Legendre polynomials and Chebyshev polynomials.

A wide range of integrals of elementary functions can be expressed using the hypergeometric function, e.g.:

${\displaystyle \int _{0}^{x}{\sqrt {1+y^{\alpha }}}\,\mathrm {d} y={\frac {x}{2+\alpha }}\left\{\alpha \;{}_{2}F_{1}\left({\tfrac {1}{\alpha }},{\tfrac {1}{2}};1+{\tfrac {1}{\alpha }};-x^{\alpha }\right)+2{\sqrt {x^{\alpha }+1}}\right\},\qquad \alpha \neq 0.}$

This occurs in connection with Mott polynomials.[5]

This occurs in the theory of Bessel functions. It provides a way to compute Bessel functions of large arguments.

${\displaystyle \operatorname {Li} _{2}(x)=\sum _{n>0}\,{x^{n}}{n^{-2}}=x\;{}_{3}F_{2}(1,1,1;2,2;x)}$ is the dilogarithm[6]

${\displaystyle Q_{n}(x;a,b,N)={}_{3}F_{2}(-n,-x,n+a+b+1;a+1,-N+1;1)}$ is a Hahn polynomial.

${\displaystyle p_{n}(t^{2})=(a+b)_{n}(a+c)_{n}(a+d)_{n}\;{}_{4}F_{3}\left({\begin{matrix}-n&a+b+c+d+n-1&a-t&a+t\\a+b&a+c&a+d\end{matrix}};1\right)}$ is a Wilson polynomial.