Generalized hypergeometric function

The generalized hypergeometric function in mathematics is a function that generalizes the Gaussian hypergeometric function and ultimately the geometric series . It belongs to the class of special functions .

The generalized hypergeometric function contains many important functions as special cases, above all the exponential function and the trigonometric functions . Indeed, there are a large number of functions that can be written as a hypergeometric function.

definition

The generalized hypergeometric function is defined by

${\displaystyle {}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z)=\sum _{k=0}^{\infty }\prod _{i=1}^{p}{\frac {\Gamma (k+a_{i})}{\Gamma (a_{i})}}\prod _{j=1}^{q}{\frac {\Gamma (b_{j})}{\Gamma (k+b_{j})}}{\frac {z^{k}}{k!}}}$,

where is the gamma function . The coefficients and the parameters are to be chosen so that the power series converge for a suitable one . ${\displaystyle \Gamma (\cdot )}$ ${\displaystyle p,q\in \mathbb {N} _{0}}$ ${\displaystyle a_{i},b_{j}\in \mathbb {C} }$ ${\displaystyle z\in \mathbb {C} }$

Another common notation of the generalized hypergeometric function is

${\displaystyle {}_{p}F_{q}\left[{\begin{matrix}a_{1},a_{2},\dots ,a_{p-1},a_{p}\\b_{1},b_{2},\dots ,b_{q-1},b_{q}\end{matrix}};z\right].}$

By choosing the coefficients and , special hypergeometric functions are finally constructed, such as the Kummersche hypergeometric function ( ) or with and the Gaussian hypergeometric function. ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p=q=1}$ ${\displaystyle p=2}$ ${\displaystyle q=1}$

Convergence conditions

Under certain conditions the power series are divergent and thus do not allow a general hypergeometric function to be represented. In particular, there are conditions for and for which the expressions or in the power series produce divergences. ${\displaystyle a_{i}}$ ${\displaystyle b_{j}}$ ${\displaystyle {\tfrac {\Gamma (k+a_{i})}{\Gamma (a_{i})}}}$ ${\displaystyle {\tfrac {\Gamma (b_{j})}{\Gamma (k+b_{j})}}}$

example 1

${\displaystyle {\begin{aligned}{}_{1}F_{1}\left(2;-1;z\right)&\;=\;\sum _{k=0}^{\infty }{\frac {\Gamma (k+2)}{\Gamma (2)}}{\frac {\Gamma (-1)}{\Gamma (k-1)}}{\frac {z^{k}}{k!}}\;=\;\sum _{k=0}^{\infty }{\frac {(k+1)!}{1}}{\frac {\Gamma (-1)}{\Gamma (k-1)}}{\frac {z^{k}}{k!}}\;=\;\sum _{k=0}^{\infty }{\frac {(k+1)}{1}}{\frac {\Gamma (-1)}{\Gamma (k-1)}}z^{k}\\&\;=\;1\;+\;2{\frac {\Gamma (-1)}{\Gamma (0)}}z\qquad \;+\;3{\frac {\Gamma (-1)}{\Gamma (1)}}z^{2}\;+\;4{\frac {\Gamma (-1)}{\Gamma (2)}}z^{3}\ \;+\;5{\frac {\Gamma (-1)}{\Gamma (3)}}z^{4}\;+\;\dotsm \\&\;=\;1\;+\;2{\frac {\Gamma (0)}{(-1)\Gamma (0)}}z\quad +\;3{\frac {\Gamma (0)}{(-1)0!}}z^{2}\;+\;4{\frac {\Gamma (0)}{(-1)1!}}z^{3}\;+\;5{\frac {\Gamma (0)}{(-1)2!}}z^{4}\;+\;\dotsm \\&\;=\;1\;-\;2z\;-\;\Gamma (0)\sum _{k=1}^{\infty }{\frac {k+2}{(k-1)!}}z^{k+1}\quad {\xrightarrow[{\lim _{x\to 0^{+}}\Gamma (x)\to \infty }]{\quad }}\quad -\infty \end{aligned}}}$

The functional equation of the gamma function with identity was used for the calculation . ${\displaystyle \Gamma (k+1)=k\Gamma (k)}$ ${\displaystyle \Gamma (-1)={\tfrac {\Gamma (0)}{-1}}}$

Example 2

${\displaystyle {\begin{aligned}{}_{0}F_{1}\left(;-1;z\right)&\;=\;\sum _{k=0}^{\infty }{\frac {\Gamma (-1)}{\Gamma (k-1)}}{\frac {z^{k}}{k!}}\\&\;=\;1\qquad +\;{\frac {\Gamma (-1)}{\Gamma (0)}}{\frac {z}{1!}}\;+\;{\frac {\Gamma (-1)}{\Gamma (1)}}{\frac {z^{2}}{2!}}\;+\;{\frac {\Gamma (-1)}{\Gamma (2)}}{\frac {z^{3}}{3!}}\;+\;{\frac {\Gamma (-1)}{\Gamma (3)}}{\frac {z^{4}}{4!}}\;+\;\dotsm \\&\;=\;1\qquad \;-\;{\frac {\Gamma (0)}{\Gamma (0)}}{\frac {z}{1!}}\quad -\;{\frac {\Gamma (0)}{\Gamma (1)}}{\frac {z^{2}}{2!}}\quad -\;{\frac {\Gamma (0)}{\Gamma (2)}}{\frac {z^{3}}{3!}}\quad -\;{\frac {\Gamma (0)}{\Gamma (3)}}{\frac {z^{4}}{4!}}\quad +\;\dotsm \\&\;=\;1\qquad \;-\;z\qquad \qquad -\;{\frac {\Gamma (0)}{0!}}{\frac {z^{2}}{2!}}\quad -\;{\frac {\Gamma (0)}{1!}}{\frac {z^{3}}{3!}}\quad -\;{\frac {\Gamma (0)}{2!}}{\frac {z^{4}}{4!}}\quad +\;\dotsm \\&\;=\;1\qquad \;-\;z\qquad \qquad -\;\Gamma (0)\sum _{k=2}^{\infty }{\frac {1}{(k-2)!}}{\frac {z^{k}}{k!}}\quad {\xrightarrow[{\lim _{x\to 0^{+}}\Gamma (x)\to \infty }]{\quad }}\quad -\infty \end{aligned}}}$

Apart from the divergences caused by the choice of parameters, the quotient criterion can be used for series :

• If is, then, according to the quotient criterion, the ratio of the coefficients is limited and possibly tends towards 0. This implies that the series converges for each finite and thus represents a whole function . An example of this is the exponential function series. ${\displaystyle p<q+1}$ ${\displaystyle z}$

• If is, then the quotient criterion shows that the ratio of the coefficients tends towards 0. This implies that the series for converges and for diverges. In order to check whether the series converges for large values ​​of , an analytical consideration is recommended. The question of convergence for is not an easy one to answer. In this case it can be shown that the series converges for absolute if: ${\displaystyle p=q+1}$ ${\displaystyle |z|<1}$ ${\displaystyle |z|>1}$ ${\displaystyle z}$ ${\displaystyle |z|=1}$ ${\displaystyle |z|=1}$

${\displaystyle \operatorname {Re} \left(\sum _{j=1}^{q}b_{j}-\sum _{i=1}^{p}a_{i}\right)>0}$.

If and is real, the following convergence condition can be given: ${\displaystyle \textstyle \sum _{i=1}^{p}a_{i}\geq \sum _{j=1}^{q}b_{j}}$ ${\displaystyle z}$

${\displaystyle \lim _{z\rightarrow 1}(1-z){\frac {\mathrm {d} \log(_{p}F_{q}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};z^{p}))}{\mathrm {d} z}}=\sum _{i=1}^{p}a_{i}-\sum _{j=1}^{q}b_{j}}$.

• If is, the quotient criterion provides an infinitely increasing ratio of the coefficients. This implies that even in the case of, the series diverges. Under these conditions a divergent or asymptotic series is obtained. On the other hand, the series can be understood as a shorthand notation for a differential equation that satisfies the sum equation. ${\displaystyle p>q+1}$ ${\displaystyle z=0}$

properties

Due to the order (degree) of the parameter and the parameter , the general hypergeometric function can be changed without changing the value of the function. So if is equal to one of the parameters , the function can be "shortened" by these two parameters, with certain exceptions for parameters with non-positive values. For example is ${\displaystyle a_{i}}$ ${\displaystyle b_{j}}$ ${\displaystyle a_{i}}$ ${\displaystyle b_{j}}$

${\displaystyle \,{}_{2}F_{1}(3,1;1;z)=\,{}_{2}F_{1}(1,3;1;z)=\,{}_{1}F_{0}(3;;z)}$.

Euler's integral transformation

The following identity makes it possible to represent the generalized hypergeometric function of higher order as an integral expression of the generalized hypergeometric function of the next lower order.

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

Differential equation

The general hypergeometric function satisfies the differential equation system:

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

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

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

The combination of these three equations results in a differential equation with : ${\displaystyle w={}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z)}$

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

Remarks:

• Differential equation (1)

${\displaystyle {\begin{aligned}&a_{i}\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{i}+1,\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{array}};z\right]\;{\stackrel {\mathrm {def} }{=}}\;\sum _{k=0}^{\infty }\prod _{i=1}^{p}{\tfrac {a_{i}\cdot \Gamma (k+a_{i}+1)}{\Gamma (a_{i}+1)}}\prod _{j=1}^{q}{\tfrac {\Gamma (b_{j})}{\Gamma (k+b_{j})}}{\tfrac {z^{k}}{k!}}\\&\qquad {\stackrel {\mathrm {def} }{=}}\;\sum _{k=0}^{\infty }{\Big (}{\tfrac {\Gamma (k+a_{1})}{\Gamma (a_{1})}}\cdots {\tfrac {\Gamma (k+a_{i-1})}{\Gamma (a_{i-1})}}\cdot {\tfrac {a_{i}\cdot \Gamma (k+a_{i}+1)}{\Gamma (a_{i}+1)}}\cdot {\tfrac {\Gamma (k+a_{i+1})}{\Gamma (a_{i+1})}}\cdots {\tfrac {\Gamma (k+a_{p})}{\Gamma (a_{p})}}{\Big )}\prod _{j=1}^{q}{\tfrac {\Gamma (b_{j})}{\Gamma (k+b_{j})}}{\tfrac {z^{k}}{k!}}\end{aligned}}}$

It should be noted that in the case of differential equation (1) the right side of the equation does not exist because the parameters did not exist and the parameters also disappear on the left side and therefore only the derivative multiplied by can be calculated. ${\displaystyle p=0}$ ${\displaystyle a_{i}}$ ${\displaystyle a_{i}}$ ${\displaystyle {\tfrac {\rm {d}}{{\rm {d}}z}}\;{}_{p}F_{q}(;b_{1},\dots ,b_{q};z)}$ ${\displaystyle z}$

• Differential equation (2)

${\displaystyle {\begin{aligned}&(b_{j}-1)\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{j}+1,\dots ,b_{q}\end{array}};z\right]\;{\stackrel {\mathrm {def} }{=}}\;\sum _{k=0}^{\infty }\prod _{i=1}^{p}{\tfrac {\Gamma (k+a_{i})}{\Gamma (a_{i})}}\prod _{j=1}^{q}{\tfrac {(b_{j}-1)\cdot \Gamma (b_{j}+1)}{\Gamma (k+b_{j}+1)}}{\tfrac {z^{k}}{k!}}\\&\qquad {\stackrel {\mathrm {def} }{=}}\;\sum _{k=0}^{\infty }\prod _{i=1}^{p}{\tfrac {\Gamma (k+a_{i})}{\Gamma (a_{i})}}{\Big (}{\tfrac {\Gamma (b_{1})}{\Gamma (k+b_{1})}}\cdots {\tfrac {\Gamma (b_{j-1})}{\Gamma (k+b_{j-1})}}\cdot {\tfrac {(b_{j}-1)\cdot \Gamma (b_{j}+1)}{\Gamma (k+b_{j}+1)}}\cdot {\tfrac {\Gamma (b_{j+1})}{\Gamma (k+b_{j+1})}}\cdots {\tfrac {\Gamma (b_{q})}{\Gamma (k+b_{q})}}{\Big )}{\tfrac {z^{k}}{k!}}\end{aligned}}}$

Here, too, it is important to note that the differential equation (2) is reduced to the shape , since the parameters do not exist. ${\displaystyle q=0}$ ${\displaystyle z{\frac {\rm {d}}{{\rm {d}}z}}{}_{p}F_{q}(;b_{1},\dots ,b_{q};z)}$ ${\displaystyle b_{j}-1}$

• Differential equation (3)

${\displaystyle {\begin{aligned}&{\frac {\prod _{i=1}^{p}a_{i}}{\prod _{j=1}^{q}b_{j}}}\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1}+1,\dots ,a_{p}+1\\b_{1}+1,\dots ,b_{q}+1\end{array}};z\right]\;{\stackrel {\mathrm {def} }{=}}\;\sum _{k=0}^{\infty }\prod _{i=1}^{p}{\tfrac {a_{i}\cdot \Gamma (k+a_{i}+1)}{\Gamma (a_{i}+1)}}\prod _{j=1}^{q}{\tfrac {\Gamma (b_{j}+1)}{b_{j}\cdot \Gamma (k+b_{j}+1)}}{\tfrac {z^{k}}{k!}}\\&\qquad =\sum _{k=0}^{\infty }{\Big (}{\tfrac {a_{1}\cdot \Gamma (k+a_{1}+1)}{\Gamma (a_{1}+1)}}\cdots {\tfrac {a_{i}\cdot \Gamma (k+a_{i}+1)}{\Gamma (a_{i}+1)}}\cdots {\tfrac {a_{p}\cdot \Gamma (k+a_{p}+1)}{\Gamma (a_{p}+1)}}{\Big )}\cdot {\Big (}{\tfrac {\Gamma (b_{1}+1)}{b_{1}\cdot \Gamma (k+b_{1}+1)}}\cdots {\tfrac {\Gamma (b_{j}+1)}{b_{j}\cdot \Gamma (k+b_{j}+1)}}\cdots {\tfrac {\Gamma (b_{q}+1)}{b_{q}\cdot \Gamma (k+b_{q}+1)}}{\Big )}{\tfrac {z^{k}}{k!}}\end{aligned}}}$

The quotient of the products for the parameters is to be interpreted in such a way that ${\displaystyle {\tfrac {\prod _{i=1}^{p}a_{i}}{\prod _{j=1}^{q}b_{j}}}}$ ${\displaystyle a_{i},b_{j}\in \mathbb {C} \setminus \{0,-1,-2,-3,\ldots \}}$

${\displaystyle \textstyle \prod _{i=1}^{p}a_{i}\;{\stackrel {\mathrm {def} }{=}}\;{\begin{cases}\prod _{i=1}^{p}a_{i}&{\text{falls}}\;p>0\\\\1&{\text{falls}}\;p=0\end{cases}}}$

and

${\displaystyle {\tfrac {1}{\prod _{j=1}^{q}b_{j}}}\;{\stackrel {\mathrm {def} }{=}}\;{\begin{cases}{\tfrac {1}{\prod _{i=1}^{q}b_{j}}}&{\text{falls}}\;q>0\\\\1&{\text{falls}}\;q=0.\end{cases}}}$

In the event that , based on the previous definition , the differential equation (3) assumes the following form ${\displaystyle p=q=0}$ ${\displaystyle {\tfrac {\prod _{i=1}^{p}a_{i}}{\prod _{j=1}^{q}b_{j}}}=1}$

${\displaystyle {\frac {\rm {d}}{{\rm {d}}z}}\;{}_{p}F_{q}(;;z)\;=\;{}_{p}F_{q}(;;z)}$

Special hypergeometric functions

The function ${\displaystyle {}_{0}F_{0}}$

→ Main article : exponential function

As indicated at the beginning, corresponds to the exponential function. The function fulfills the differential equation: ${\displaystyle {}_{0}F_{0}(;;z)=\mathrm {e} ^{z}}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}w=w}$

proof

${\displaystyle {}_{0}F_{0}\left(;;z\right)=\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}=\mathrm {e} ^{z}}$

The function ${\displaystyle {}_{0}F_{1}}$

The function of the type is the so-called confluent hypergeometric limit function . The series satisfies the differential equation: ${\displaystyle {}_{0}F_{1}(;a;z)}$

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

It is closely related to the Bessel functions :

${\displaystyle {}_{0}F_{1}\left(;1+a;-{\frac {z^{2}}{4}}\right)=\Gamma (a+1)\cdot \left({\frac {z}{2}}\right)^{-a}\cdot J_{a}(z)\quad }$wherein the Bessel function is ${\displaystyle J_{a}(z)}$

${\displaystyle {}_{0}F_{1}\left(;1+a;{\frac {z^{2}}{4}}\right)=\Gamma (a+1)\left({\frac {z}{2}}\right)^{-a}\cdot I_{a}(z)\quad }$with as a modified armchair function ${\displaystyle I_{a}(z)=e^{-i{\frac {\pi }{2}}a}J_{a}(z)}$

Derived functions of the series are for example:

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

or

${\displaystyle {}_{0}F_{1}\left(;{\frac {3}{2}};-{\frac {z^{2}}{4}}\right)={\frac {\sin z}{z}}}$.

example

The cosine function should be considered :

${\displaystyle {\begin{aligned}{}_{0}F_{1}\left(;{\tfrac {1}{2}};-{\tfrac {z^{2}}{4}}\right)&=\sum _{k=0}^{\infty }{\frac {\Gamma ({\frac {1}{2}})}{\Gamma (k+{\frac {1}{2}})}}{\frac {(-{\frac {z^{2}}{4}})^{k}}{k!}}\\&={\frac {\Gamma ({\frac {1}{2}})}{\Gamma ({\frac {1}{2}})}}{\frac {(-{\frac {z^{2}}{4}})^{0}}{0!}}\;\;\;+{\frac {\Gamma ({\frac {1}{2}})}{\Gamma ({\frac {3}{2}})}}{\frac {-{\frac {z^{2}}{4}}}{1}}&&+{\frac {\Gamma ({\frac {1}{2}})}{\Gamma ({\frac {5}{2}})}}{\frac {(-{\frac {z^{2}}{4}})^{2}}{2}}&&+{\frac {\Gamma ({\frac {1}{2}})}{\Gamma ({\frac {7}{2}})}}{\frac {(-{\frac {z^{2}}{4}})^{3}}{2\cdot 3}}&+\cdots \\&={\frac {\Gamma ({\frac {1}{2}})}{\Gamma ({\frac {1}{2}})}}{\frac {1}{1}}\qquad \quad \ \,+{\frac {\Gamma ({\frac {1}{2}})}{{\frac {1}{2}}\Gamma ({\frac {1}{2}})}}{\frac {-z^{2}}{4}}&&+{\frac {\Gamma ({\frac {1}{2}})}{{\frac {3}{2}}{\frac {1}{2}}\Gamma ({\frac {1}{2}})}}{\frac {z^{4}}{4^{2}\cdot 2!}}&&+{\frac {\Gamma ({\frac {1}{2}})}{{\frac {5}{2}}{\frac {3}{2}}{\frac {1}{2}}\Gamma ({\frac {1}{2}})}}{\frac {-z^{6}}{4^{3}\cdot 3!}}&+\cdots \end{aligned}}}$

Here we used that is and thus , etc. As you can see, the terms are abbreviated everywhere; the remaining fractions can easily be summed up ${\displaystyle \Gamma (x+1)=x\Gamma (x)}$ ${\displaystyle \Gamma ({\tfrac {3}{2}})={\tfrac {1}{2}}\Gamma ({\tfrac {1}{2}})}$ ${\displaystyle \Gamma ({\tfrac {1}{2}})}$

${\displaystyle {}_{0}F_{1}(;{\tfrac {1}{2}};-{\tfrac {z^{2}}{4}})=1-{\frac {z^{2}}{2!}}+{\frac {z^{4}}{4!}}-{\frac {z^{6}}{6!}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k)!}}=\cos z}$

The function ${\displaystyle {}_{1}F_{0}}$

The differential equation also fulfills directly as an elementary function : ${\displaystyle {}_{1}F_{0}(a;;z)=(1-z)^{-a}}$

${\displaystyle (1-z){\frac {\mathrm {d} w}{\mathrm {d} z}}=aw}$

proof

${\displaystyle {\begin{aligned}{}_{1}F_{0}\left(a;;z\right)&=\quad \sum _{k=0}^{\infty }{\frac {\Gamma (k+a)}{\Gamma (a)}}{\frac {z^{k}}{k!}}&&=\quad \sum _{k=0}^{\infty }{\frac {(k+a-1)!}{(a-1)!}}{\frac {z^{k}}{k!}}&&=\quad \sum _{k=0}^{\infty }{\frac {(k+a-1)!}{(a+k-1-k)!}}{\frac {z^{k}}{k!}}\\&=\quad \sum _{k=0}^{\infty }{\binom {a+k-1}{k}}z^{k}&&=\quad \sum _{k=0}^{\infty }{\binom {-a}{k}}(-1)^{k}z^{k}&&=\quad \sum _{k=0}^{\infty }{\binom {-a}{k}}(-z)^{k}\\&=\quad (1-z)^{-a}\end{aligned}}}$

Here the binomial coefficient was used in analysis with identity . The result is the binomial series . ${\displaystyle {\tbinom {-a}{k}}=(-1)^{k}{\tbinom {a+k-1}{k}}}$

The function ${\displaystyle {}_{1}F_{1}}$

The function is called Kummer's function (after Ernst Eduard Kummer ). It is often referred to as the confluent hypergeometric series and satisfies Kummer's differential equation: ${\displaystyle {}_{1}F_{1}(a;b;z)}$

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

Derived functions are for example:

${\displaystyle {}_{1}F_{1}\left(a;a+1;-z\right)=az^{-a}\gamma (a,z)}$wherein the incomplete gamma function is ${\displaystyle \gamma (a,z)}$

or

${\displaystyle {}_{1}F_{1}\left(1;a+1;z\right)=az^{-a}e^{z}\gamma (a,z)}$

The function ${\displaystyle {}_{2}F_{0}}$

The function appears in connection with the integral exponential function . ${\displaystyle Ei(z)}$

The function ${\displaystyle {}_{2}F_{1}}$

→ Main article : Gaussian hypergeometric function

Historically most important is the hypergeometric function . It is also referred to as the Gaussian hypergeometric function , ordinary hypergeometric function, or often just as hypergeometric function. To differentiate, generalized hypergeometric function is used for the designation , otherwise there is a risk of confusion. The function was the first to be fully investigated by Carl Friedrich Gauß , particularly with regard to convergence. It satisfies the differential equation ${\displaystyle {}_{2}F_{1}(a,b;c;z)}$ ${\displaystyle {}_{p}F_{q}}$

${\displaystyle z(1-z){\frac {\mathrm {d} ^{2}w}{\mathrm {d} z^{2}}}+(c-(a+b+1)z){\frac {\mathrm {d} w}{\mathrm {d} z}}-abw=0}$,

which is called the hypergeometric differential equation .

The function ${\displaystyle {}_{3}F_{0}}$

The function appears in connection with the Mott polynomial .

The function ${\displaystyle {}_{3}F_{1}}$

The function appears in connection with the armchair function .

Further generalizations

The generalized hypergeometric function can be generalized even further by introducing prefactors before the , thus further increasing the complexity of the function. Just to modify the sign of , two further indices would be necessary: ${\displaystyle k}$ ${\displaystyle k}$

${\displaystyle {\begin{aligned}&F_{pqrs}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};c_{1},\dots ,c_{r};d_{1},\dots ,d_{s};z)\\&\qquad \qquad =\sum _{k=0}^{\infty }\prod _{i=1}^{p}{\frac {\Gamma (k+a_{i})}{\Gamma (a_{i})}}\prod _{j=1}^{q}{\frac {\Gamma (-k+b_{j})}{\Gamma (b_{j})}}\prod _{l=1}^{r}{\frac {\Gamma (c_{l})}{\Gamma (k+c_{l})}}\prod _{m=1}^{s}{\frac {\Gamma (d_{m})}{\Gamma (-k+d_{m})}}{\frac {z^{k}}{k!}}\end{aligned}}}$

If these prefactors are not necessarily whole numbers, the Fox – Wright functions are obtained as a generalization .

literature

• Eduard Heine : Handbuch der Kugelfunctionen p. 91 Georg Reimer, Berlin, 1861.
• Felix Klein : Lectures on the hypergeometric function Springer, Berlin, reprint 1981.
• Ludwig Bieberbach : Theory of differential equations Springer, Berlin, 1930.

Individual evidence

1. ↑ J. Quigley, KJ Wilson, L. Walls, T. Bedford: A Bayes linear Bayes Method for Estimation of Correlated Event Rates In: Risk Analysis 2013 doi = 10.1111 / risa.12035
2. ↑ Lucy Joan Slater, "Generalized Hypergeometric Functions" In: "Cambridge University Press." 1966 ISBN 0-521-06483-X (a reprint was published as a paperback in 2008: ISBN 978-0-521-09061-2 )