Generalized hypergeometric function#The series 0F1

A hypergeometric series is formally defined as a power series

:$\backslash beta\_0\; +\; \backslash beta\_1\; z\; +\; \backslash beta\_2\; z^2\; +\; \backslash dots\; =\; \backslash sum\_\{n\; \backslash geqslant\; 0\}\; \backslash beta\_n\; z^n$

in which the ratio of successive coefficients is a rational function of n. That is,

:$\backslash frac\{\backslash beta\_\{n+1\}\}\{\backslash beta\_n\}\; =\; \backslash frac\{A(n)\}\{B(n)\}$

where A(n) and B(n) are polynomials in n.

For example, in the case of the series for the exponential function,

:$1+\backslash frac\{z\}\{1!\}+\backslash frac\{z^2\}\{2!\}+\backslash frac\{z^3\}\{3!\}+\backslash cdots,$

we have:

:$\backslash beta\_n\; =\; \backslash frac\{1\}\{n!\},\; \backslash qquad\; \backslash frac\{\backslash beta\_\{n+1\}\}\{\backslash beta\_n\}\; =\; \backslash frac\{1\}\{n+1\}.$

So this satisfies the definition with {{math|A(n) {{=}} 1}} and {{math|B(n) {{=}} n + 1}}.

It is customary to factor out the leading term, so β₀ is assumed to be 1. The polynomials can be factored into linear factors of the form (a_j + n) and (b_k + n) respectively, where the a_j and b_k are complex numbers.

For historical reasons, it is assumed that (1 + n) is a factor of B. If this is not already the case then both A and B can be multiplied by this factor; the factor cancels so the terms are unchanged and there is no loss of generality.

The ratio between consecutive coefficients now has the form

:$\backslash frac\{c(a\_1+n)\backslash cdots(a\_p+n)\}\{d(b\_1+n)\backslash cdots(b\_q+n)(1+n)\}$,

where c and d are the leading coefficients of A and B. The series then has the form

:$1\; +\; \backslash frac\{a\_1\backslash cdots\; a\_p\}\{b\_1\backslash cdots\; b\_q\; \backslash cdot\; 1\}\backslash frac\{cz\}\{d\}\; +\; \backslash frac\{a\_1\backslash cdots\; a\_p\}\{b\_1\backslash cdots\; b\_q\; \backslash cdot\; 1\}\; \backslash frac\{(a\_1+1)\backslash cdots(a\_p+1)\}\{(b\_1+1)\backslash cdots\; (b\_q+1)\; \backslash cdot\; 2\}\backslash left(\backslash frac\{cz\}\{d\}\backslash right)^2+\backslash cdots$,

or, by scaling z by the appropriate factor and rearranging,

:$1\; +\; \backslash frac\{a\_1\backslash cdots\; a\_p\}\{b\_1\backslash cdots\; b\_q\}\backslash frac\{z\}\{1!\}\; +\; \backslash frac\{a\_1(a\_1+1)\backslash cdots\; a\_p(a\_p+1)\}\{b\_1(b\_1+1)\backslash cdots\; b\_q(b\_q+1)\}\backslash frac\{z^2\}\{2!\}+\backslash cdots$.

This has the form of an exponential generating function. This series is usually denoted by

:$\{\}\_pF\_q(a\_1,\backslash ldots,a\_p;b\_1,\backslash ldots,b\_q;z)$

or

:

Using the rising factorial or Pochhammer symbol

:$\backslash begin\{align\}$

(a)_0 &= 1, \\

(a)_n &= a(a+1)(a+2) \cdots (a+n-1), && n \geq 1

\end{align}

this can be written

:$\backslash ,\{\}\_pF\_q(a\_1,\backslash ldots,a\_p;b\_1,\backslash ldots,b\_q;z)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(a\_1)\_n\backslash cdots(a\_p)\_n\}\{(b\_1)\_n\backslash cdots(b\_q)\_n\}\; \backslash ,\; \backslash frac\; \{z^n\}\; \{n!\}.$

(Note that this use of the Pochhammer symbol is not standard; however it is the standard usage in this context.)

When all the terms of the series are defined and it has a non-zero radius of convergence, then the series defines an analytic function. Such a function, and its analytic continuations, is called the hypergeometric function.

The case when the radius of convergence is 0 yields many interesting series in mathematics, for example the incomplete gamma function has the asymptotic expansion

:$\backslash Gamma(a,z)\; \backslash sim\; z^\{a-1\}e^\{-z\}\backslash left(1+\backslash frac\{a-1\}\{z\}+\backslash frac\{(a-1)(a-2)\}\{z^2\}\; +\; \backslash cdots\backslash right)$

which could be written z^a−1e^−z ₂F₀(1−a,1;;−z⁻¹). However, the use of the term hypergeometric series is usually restricted to the case where the series defines an actual analytic function.

The ordinary hypergeometric series should not be confused with the basic hypergeometric series, which, despite its name, is a rather more complicated and recondite series. The "basic" series is the q-analog of the ordinary hypergeometric series. There are several such generalizations of the ordinary hypergeometric series, including the ones coming from zonal spherical functions on Riemannian symmetric spaces.

The series without the factor of n! in the denominator (summed over all integers n, including negative) is called the bilateral hypergeometric series.

There are certain values of the a_j and b_k for which the numerator or the denominator of the coefficients is 0.

• If any a_j is a non-positive integer (0, −1, −2, etc.) then the series only has a finite number of terms and is, in fact, a polynomial of degree −a_j.
• If any b_k is a non-positive integer (excepting the previous case with b_k < a_j) then the denominators become 0 and the series is undefined.

Excluding these cases, the ratio test can be applied to determine the radius of convergence.

• If p < q + 1 then the ratio of coefficients tends to zero. This implies that the series converges for any finite value of z and thus defines an entire function of z. An example is the power series for the exponential function.
• If p = q + 1 then the ratio of coefficients tends to one. This implies that the series converges for |z| < 1 and diverges for |z| > 1. Whether it converges for |z| = 1 is more difficult to determine. Analytic continuation can be employed for larger values of z.
• If p > q + 1 then the ratio of coefficients grows without bound. This implies that, besides z = 0, the series diverges. This is then a divergent or asymptotic series, or it can be interpreted as a symbolic shorthand for a differential equation that the sum satisfies formally.

The question of convergence for p=q+1 when z is on the unit circle is more difficult. It can be shown that the series converges absolutely at z = 1 if

:$\backslash Re\backslash left(\backslash sum\; b\_k\; -\; \backslash sum\; a\_j\backslash right)>0$.

Further, if p=q+1, $\backslash sum\_\{i=1\}^\{p\}a\_\{i\}\backslash geq\backslash sum\_\{j=1\}^\{q\}b\_\{j\}$ and z is real, then the following convergence result holds {{harvtxt|Quigley|Wilson|Walls|Bedford|2013}}:

:$\backslash lim\_\{z\backslash rightarrow\; 1\}(1-z)\backslash frac\{d\backslash log(\_\{p\}F\_\{q\}(a\_\{1\},\backslash ldots,a\_\{p\};b\_\{1\},\backslash ldots,b\_\{q\};z^\{p\}))\}\{dz\}=\backslash sum\_\{i=1\}^\{p\}a\_\{i\}-\backslash sum\_\{j=1\}^\{q\}b\_\{j\}$.

It is immediate from the definition that the order of the parameters a_j, or the order of the parameters b_k can be changed without changing the value of the function. Also, if any of the parameters a_j is equal to any of the parameters b_k, then the matching parameters can be "cancelled out", with certain exceptions when the parameters are non-positive integers. For example,

:$\backslash ,\{\}\_2F\_1(3,1;1;z)\; =\; \backslash ,\{\}\_2F\_1(1,3;1;z)\; =\; \backslash ,\{\}\_1F\_0(3;;z)$.

This cancelling is a special case of a reduction formula that may be applied whenever a parameter on the top row differs from one on the bottom row by a non-negative integer.{{cite book|first1=A. P.|last1=Prudnikov|first2=Yu. A.|last2=Brychkov|first3=O. I.|last3=Marichev|title=Integrals & Series Volume 3: More Special Functions|publisher=Gordon and Breach|year=1990|pages=439}}{{cite journal|first1=Per W. |last1=Karlsson|title = Hypergeometric functions with integral parameter differences|year=1970 | journal=J. Math. Phys.|volume=12|number=2|doi=10.1063/1.1665587|pages=270–271|url=https://backend.orbit.dtu.dk/ws/files/3645014/Per.pdf }}

:$$

{}_{A+1}F_{B+1}\left[

\begin{array}{c}

a_{1},\ldots ,a_{A},c + n \\

b_{1},\ldots ,b_{B},c

\end{array}

;z\right] = \sum_{j = 0}^n \binom{n}{j} \frac{z^j}{(c)_j} \frac{\prod_{i = 1}^A (a_i)_j}{\prod_{i = 1}^B (b_i)_j} {}_AF_B\left[

\begin{array}{c}

a_{1} + j,\ldots ,a_{A} + j \\

b_{1} + j,\ldots ,b_{B} + j

\end{array}

;z\right]

The following basic identity is very useful as it relates the higher-order hypergeometric functions in terms of integrals over the lower order ones{{harv|Slater|1966|loc=Equation (4.1.2)}}

:$\{\}\_\{A+1\}F\_\{B+1\}\backslash left[$

\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

:$\backslash begin\{align\}$

\left (z\frac{{\rm{d}}}{{\rm{d}}z} + a_j \right ){}_pF_q\left[ \begin{array}{c} a_1,\dots,a_j,\dots,a_p \\ b_1,\dots,b_q\end{array} ;z\right] &= a_j \; {}_pF_q\left[ \begin{array}{c} a_1,\dots,a_j+1,\dots,a_p \\ b_1,\dots,b_q \end{array} ;z\right] \\

\end{align}

and

$\backslash begin\{align\}$

\left (z\frac{{\rm{d}}}{{\rm{d}}z} + b_k - 1 \right ){}_pF_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) \; {}_pF_q\left[ \begin{array}{c} a_1,\dots,a_p \\ b_1,\dots,b_k-1,\dots,b_q \end{array} ;z \right] \text{ for } b_k\neq 1

\end{align}

Additionally,

$\backslash begin\{align\}$

\frac{{\rm{d}}}{{\rm{d}}z} \; {}_pF_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}\; {}_pF_q\left[ \begin{array}{c} a_1+1,\dots,a_p+1 \\ b_1+1,\dots,b_q+1 \end{array} ;z \right]

\end{align}

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

:$z\backslash prod\_\{n=1\}^\{p\}\backslash left(z\backslash frac\{\{\backslash rm\{d\}\}\}\{\{\backslash rm\{d\}\}z\}\; +\; a\_n\backslash right)w\; =\; z\backslash frac\{\{\backslash rm\{d\}\}\}\{\{\backslash rm\{d\}\}z\}\backslash prod\_\{n=1\}^\{q\}\backslash left(z\backslash frac\{\{\backslash rm\{d\}\}\}\{\{\backslash rm\{d\}\}z\}\; +\; b\_n-1\backslash right)w$.

Take the following operator:

:$\backslash vartheta\; =\; z\backslash frac\{\{\backslash rm\{d\}\}\}\{\{\backslash rm\{d\}\}z\}.$

From the differentiation formulas given above, the linear space spanned by

:$\{\}\_pF\_q(a\_1,\backslash dots,a\_p;b\_1,\backslash dots,b\_q;z),\; \backslash vartheta\backslash ;\; \{\}\_pF\_q(a\_1,\backslash dots,a\_p;b\_1,\backslash dots,b\_q;z)$

contains each of

:$\{\}\_pF\_q(a\_1,\backslash dots,a\_j+1,\backslash dots,a\_p;b\_1,\backslash dots,b\_q;z),$

:$\{\}\_pF\_q(a\_1,\backslash dots,a\_p;b\_1,\backslash dots,b\_k-1,\backslash dots,b\_q;z),$

:$z\backslash ;\; \{\}\_pF\_q(a\_1+1,\backslash dots,a\_p+1;b\_1+1,\backslash dots,b\_q+1;z),$

:$\{\}\_pF\_q(a\_1,\backslash dots,a\_p;b\_1,\backslash dots,b\_q;z).$

Since the space has dimension 2, any three of these p+q+2 functions are linearly dependent:

{{cite journal|first1=J. E. |last1=Gottschalk|first2=E. N.|last2=Maslen|title=Reduction formulae for generalised hypergeometric functions of one variable| journal=J. Phys. A: Math. Gen.|volume=21|pages=1983–1998|year=1988|issue=9 |doi=10.1088/0305-4470/21/9/015|bibcode=1988JPhA...21.1983G }}

{{cite journal|first1=D. |last1=Rainville|title=The contiguous function relations for pFq with application to Bateman's J and Rice's H| journal=Bull. Amer. Math. Soc.|volume=51|number=10|pages=714–723|year=1945|doi=10.1090/S0002-9904-1945-08425-0|doi-access=free}}

:$$

(a_i-b_j+1){}_pF_q(...a_i..;...,b_j...;z)

=

a_i\,{}_pF_q(...a_i+1..;...,b_j...;z)

-(b_j-1){}_pF_q(...a_i..;...,b_j-1...;z).

:$$

(a_i-a_j){}_pF_q(...a_i..a_j..;.....;z)

=

a_i\,{}_pF_q(...a_i+1..a_j..;......;z)

-a_j\,{}_pF_q(...a_i..a_j+1...;....;z).

:$$

b_j\,{}_pF_q(...a_i....;..b_j...;z)

=

a_i\,{}_pF_q(...a_i+1....;..b_j+1...;z)

+(b_j-a_i){}_pF_q(...a_i....;..b_j+1...;z)

.

:$$

(a_i-1){}_pF_q(...a_i..a_j;...;z)

=

(a_i-a_j-1){}_pF_q(...a_i-1..a_j;...;z)

+a_j\,{}_pF_q(...a_i-1..a_j+1;...;z)

.

These dependencies can be written out to generate a large number of identities involving $\{\}\_pF\_q$.

For example, in the simplest non-trivial case,

:$\backslash ;\; \{\}\_0F\_1(;a;z)\; =\; (1)\; \backslash ;\; \{\}\_0F\_1(;a;z)$,

:$\backslash ;\; \{\}\_0F\_1(;a-1;z)\; =\; (\backslash frac\{\backslash vartheta\}\{a-1\}+1)\; \backslash ;\; \{\}\_0F\_1(;a;z)$,

:$z\; \backslash ;\; \{\}\_0F\_1(;a+1;z)\; =\; (a\backslash vartheta)\; \backslash ;\; \{\}\_0F\_1(;a;z)$,

So

:$\backslash ;\; \{\}\_0F\_1(;a-1;z)-\; \backslash ;\; \{\}\_0F\_1(;a;z)\; =\; \backslash frac\{z\}\{a(a-1)\}\; \backslash ;\; \{\}\_0F\_1(;a+1;z)$.

This, and other important examples,

:$\backslash ;\; \{\}\_1F\_1(a+1;b;z)-\; \backslash ,\; \{\}\_1F\_1(a;b;z)\; =\; \backslash frac\{z\}\{b\}\; \backslash ;\; \{\}\_1F\_1(a+1;b+1;z)$,

:$\backslash ;\; \{\}\_1F\_1(a;b-1;z)-\; \backslash ,\; \{\}\_1F\_1(a;b;z)\; =\; \backslash frac\{az\}\{b(b-1)\}\; \backslash ;\; \{\}\_1F\_1(a+1;b+1;z)$,

:$\backslash ;\; \{\}\_1F\_1(a;b-1;z)-\; \backslash ,\; \{\}\_1F\_1(a+1;b;z)\; =\; \backslash frac\{(a-b+1)z\}\{b(b-1)\}\; \backslash ;\; \{\}\_1F\_1(a+1;b+1;z)$

:$\backslash ;\; \{\}\_2F\_1(a+1,b;c;z)-\; \backslash ,\; \{\}\_2F\_1(a,b;c;z)\; =\; \backslash frac\{bz\}\{c\}\; \backslash ;\; \{\}\_2F\_1(a+1,b+1;c+1;z)$,

:$\backslash ;\; \{\}\_2F\_1(a+1,b;c;z)-\; \backslash ,\; \{\}\_2F\_1(a,b+1;c;z)\; =\; \backslash frac\{(b-a)z\}\{c\}\; \backslash ;\; \{\}\_2F\_1(a+1,b+1;c+1;z)$,

:$\backslash ;\; \{\}\_2F\_1(a,b;c-1;z)-\; \backslash ,\; \{\}\_2F\_1(a+1,b;c;z)\; =\; \backslash frac\{(a-c+1)bz\}\{c(c-1)\}\; \backslash ;\; \{\}\_2F\_1(a+1,b+1;c+1;z)$,

can be used to generate continued fraction expressions known as Gauss's continued fraction.

Similarly, by applying the differentiation formulas twice, there are $\backslash binom\{p+q+3\}\{2\}$ such functions contained in

:$\backslash \{1,\; \backslash vartheta,\; \backslash vartheta^2\backslash \}\backslash ;\; \{\}\_p\; F\_q\; (a\_1,\backslash dots,a\_p;b\_1,\backslash dots,b\_q;z),$

which has dimension three so any four are linearly dependent. This generates more identities and the process can be continued. The identities thus generated can be combined with each other to produce new ones in a different way.

A function obtained by adding ±1 to exactly one of the parameters a_j, b_k in

:$\{\}\_pF\_q(a\_1,\backslash dots,a\_p;b\_1,\backslash dots,b\_q;z)$

is called contiguous to

:$\{\}\_pF\_q(a\_1,\backslash dots,a\_p;b\_1,\backslash dots,b\_q;z).$

Using the technique outlined above, an identity relating $\{\}\_0F\_1(;a;z)$ and its two contiguous functions can be given, six identities relating $\{\}\_1F\_1(a;b;z)$ and any two of its four contiguous functions, and fifteen identities relating $\{\}\_2F\_1(a,b;c;z)$ and any two of its six contiguous functions have been found. The first one was derived in the previous paragraph. The last fifteen were given by {{Harvard citation|Gauss|1813}}.

{{for|identities involving the Gauss hypergeometric function ₂F₁|Hypergeometric function}}

A number of other hypergeometric function identities were discovered in the nineteenth and twentieth centuries. A 20th century contribution to the methodology of proving these identities is the Egorychev method.

Saalschütz's theoremSee {{harv|Slater|1966|loc=Section 2.3.1}} or {{harv|Bailey|1935|loc=Section 2.2}} for a proof, or the ProofWiki. {{harv|Saalschütz|1890}} is

:$\{\}\_3F\_2\; (a,b,\; -n;c,\; 1+a+b-c-n;1)=\; \backslash frac\{(c-a)\_n(c-b)\_n\}\{(c)\_n(c-a-b)\_n\}.$

For extension of this theorem, see a research paper by Rakha & Rathie. According to {{Harvard citation|Andrews|Askey|Roy|1999|p=69}}, it was in fact first discovered by Pfaff in 1797.Pfaff, J. F. [1797]. Observations analyticae ad L. Euleri Institutiones Calculi Integralis.  Vol. IV, Supplem. II et IV, Historie de 1793, Nova Acata Acad. Scie. Petropolitanae.  XI, 38-57. (Note: The history section is paged separately from the scientific section  of this journal.)

{{Main|Dixon's identity}}

Dixon's identity,See {{harv|Bailey|1935|loc=Section 3.1}} for a detailed proof. An alternative proof is in {{harv|Slater|1966|loc=Section 2.3.3}} first proved by {{harvtxt|Dixon|1902}}, gives the sum of a well-poised ₃F₂ at 1:

:$\{\}\_3F\_2\; (a,b,c;1+a-b,1+a-c;1)=\; \backslash frac\{\backslash Gamma(1+\backslash frac\{a\}\{2\})\backslash Gamma(1+\backslash frac\{a\}\{2\}-b-c)\backslash Gamma(1+a-b)\backslash Gamma(1+a-c)\}\{\backslash Gamma(1+a)\backslash Gamma(1+a-b-c)\backslash Gamma(1+\backslash frac\{a\}\{2\}-b)\backslash Gamma(1+\backslash frac\{a\}\{2\}-c)\}.$

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

Dougall's formula {{harvs|authorlink=John Dougall (mathematician)|first=|last=Dougall|year=1907}} gives the sum of a very well-poised series that is terminating and 2-balanced.

:$\backslash begin\{align\}$

{}_7F_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{align}

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

:$1+2a=b+c+d+e-m.$

Many of the other formulas for special values of hypergeometric functions can be derived from this as special or limiting cases. It is also called the Dougall-Ramanujan identity. It is a special case of Jackson's identity, and it gives Dixon's identity and Saalschütz's theorem as special cases.{{Cite web |last=Weisstein |first=Eric W. |title=Dougall-Ramanujan Identity |url=https://mathworld.wolfram.com/Dougall-RamanujanIdentity.html |access-date=2025-03-13 |website=mathworld.wolfram.com |language=en}}

Identity 1.

:$e^\{-x\}\; \backslash ;\; \{\}\_2F\_2(a,1+d;c,d;x)=\; \{\}\_2F\_2(c-a-1,f+1;c,f;-x)$

where

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

Identity 2.

:$e^\{-\backslash frac\; x\; 2\}\; \backslash ,\; \{\}\_2F\_2\; \backslash left(a,\; 1+b;\; 2a+1,\; b;\; x\backslash right)=\; \{\}\_0F\_1\; \backslash left(;a+\backslash tfrac\{1\}\{2\};\; \backslash tfrac\; \{x^2\}\; \{16\}\backslash right)\; -\; \backslash frac\{x\backslash left(1-\backslash tfrac\{2a\}\; b\backslash right)\}\{2(2a+1)\}\backslash ;\; \{\}\_0F\_1\; \backslash left(;a+\backslash tfrac\; 3\; 2;\; \backslash tfrac\; \{x^2\}\; \{16\}\backslash right),$

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

Identity 3.

:$e^\{-\backslash frac\; x\; 2\}\; \backslash ,\; \{\}\_1F\_1(a,2a,x)=\; \{\}\_0F\_1\; \backslash left\; (;a+\backslash tfrac\; 1\; 2;\; \backslash tfrac\{x^2\}\{16\}\; \backslash right\; )$.

Identity 4.

:$\backslash begin\{align\}$

{}_2F_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}} \; {}_1F_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}} \; {}_1F_1(c-a;c+i;-x)\frac{x^i}{i!},

\end{align}

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

Kummer's relation is

:$\{\}\_2F\_1\backslash left(2a,2b;a+b+\backslash tfrac\; 1\; 2;x\backslash right)=\; \{\}\_2F\_1\backslash left(a,b;\; a+b+\backslash tfrac\; 1\; 2;\; 4x(1-x)\backslash right).$

{{Main|Clausen's formula}}

Clausen's formula

:$\{\}\_3F\_2(2c-2s-1,\; 2s,\; c-\backslash tfrac\; 1\; 2;\; 2c-1,\; c;\; x)=\backslash ,\; \{\}\_2F\_1(c-s-\backslash tfrac\; 1\; 2,s;\; c;\; x)^2$

was used by de Branges to prove the Bieberbach conjecture.

Many of the special functions in mathematics are special cases of the confluent hypergeometric function or the hypergeometric function; see the corresponding articles for examples.

{{Main|Exponential function}}

As noted earlier, $\{\}\_0F\_0(;;z)\; =\; e^z$. The differential equation for this function is $\backslash frac\{d\}\{dz\}w\; =\; w$, which has solutions $w\; =\; ke^z$ where k is a constant.

The functions of the form $\{\}\_0F\_1(;a;z)$ are called confluent hypergeometric limit functions and are closely related to Bessel functions.

The relationship is:

:$J\_\backslash alpha(x)=\backslash frac\{(\backslash tfrac\{x\}\{2\})^\backslash alpha\}\{\backslash Gamma(\backslash alpha+1)\}\; \{\}\_0F\_1\backslash left\; (;\backslash alpha+1;\; -\backslash tfrac\{1\}\{4\}x^2\; \backslash right\; ).$

:$I\_\backslash alpha(x)=\backslash frac\{(\backslash tfrac\{x\}\{2\})^\backslash alpha\}\{\backslash Gamma(\backslash alpha+1)\}\; \{\}\_0F\_1\backslash left\; (;\backslash alpha+1;\; \backslash tfrac\{1\}\{4\}x^2\; \backslash right\; ).$

The differential equation for this function is

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

or

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

When a is not a positive integer, the substitution

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

gives a linearly independent solution

:$z^\{1-a\}\backslash ;\{\}\_0F\_1(;2-a;z),$

so the general solution is

:$k\backslash ;\{\}\_0F\_1(;a;z)+l\; z^\{1-a\}\backslash ;\{\}\_0F\_1(;2-a;z)$

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.)

A special case is:

:$\{\}\_0F\_1\backslash left(;\backslash frac\{1\}\{2\};-\backslash frac\{z^2\}\{4\}\backslash right)=\backslash cos\; z$

{{Main|Binomial series}}

An important case is:

:$\{\}\_1F\_0(a;;z)\; =\; (1-z)^\{-a\}.$

The differential equation for this function is

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

or

:$(1-z)\backslash frac\{dw\}\{dz\}\; =\; aw,$

which has solutions

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

where k is a constant.

:$\{\}\_1F\_0(1;;z)\; =\; \backslash sum\_\{n\; \backslash geqslant\; 0\}\; z^n\; =\; (1-z)^\{-1\}$ is the geometric series with ratio z and coefficient 1.

:$z\; ~\; \{\}\_1F\_0(2;;z)\; =\; \backslash sum\_\{n\; \backslash geqslant\; 0\}\; n\; z^n\; =\; z\; (1-z)^\{-2\}$ is also useful.

{{main|Confluent hypergeometric function}}

The functions of the form $\{\}\_1F\_1(a;b;z)$ are called confluent hypergeometric functions of the first kind, also written $M(a;b;z)$. The incomplete gamma function $\backslash gamma(a,z)$ is a special case.

The differential equation for this function is

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

or

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

When b is not a positive integer, the substitution

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

gives a linearly independent solution

:$z^\{1-b\}\backslash ;\{\}\_1F\_1(1+a-b;2-b;z),$

so the general solution is

:$k\backslash ;\{\}\_1F\_1(a;b;z)+l\; z^\{1-b\}\backslash ;\{\}\_1F\_1(1+a-b;2-b;z)$

where k, l are constants.

When a is a non-positive integer, −n, $\{\}\_1F\_1(-n;b;z)$ 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.

Relations to other functions are known for certain parameter combinations only.

The function $x\backslash ;\; \{\}\_1F\_2\backslash left(\backslash frac\{1\}\{2\};\backslash frac\{3\}\{2\},\backslash frac\{3\}\{2\};-\backslash frac\{x^2\}\{4\}\backslash right)$ is the antiderivative of the cardinal sine. With modified values of $a\_1$ and $b\_1$, one obtains the antiderivative of $\backslash sin(x^\backslash beta)/x^\backslash alpha$.Victor Nijimbere, Ural Math J vol 3(1) and https://arxiv.org/abs/1703.01907 (2017)

The Lommel function is $s\_\{\backslash mu,\; \backslash nu\}\; (z)\; =\; \backslash frac\{z^\{\backslash mu\; +\; 1\}\}\{(\backslash mu\; -\; \backslash nu\; +\; 1)(\backslash mu\; +\; \backslash nu\; +\; 1)\}\; \{\}\_1F\_2(1;\; \backslash frac\{\backslash mu\}\{2\}\; -\; \backslash frac\{\backslash nu\}\{2\}\; +\; \backslash frac\{3\}\{2\}\; ,\; \backslash frac\{\backslash mu\}\{2\}\; +\; \backslash frac\{\backslash nu\}\{2\}\; +\; \backslash frac\{3\}\{2\}\; ;-\backslash frac\{z^2\}\{4\})$.Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)

The confluent hypergeometric function of the second kind can be written as:{{cite web |title=DLMF: §13.6 Relations to Other Functions ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions |url=https://dlmf.nist.gov/13.6 |website=dlmf.nist.gov}}

:$U(a,b,z)\; =\; z^\{-a\}\; \backslash ;\; \{\}\_2\; F\_0\; \backslash left(\; a,\; a-b+1;\; ;\; -\backslash frac\{1\}\{z\}\backslash right).$

{{main|Hypergeometric function}}

Historically, the most important are the functions of the form $\{\}\_2F\_1(a,b;c;z)$. 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

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

or

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

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

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

gives a linearly independent solution

:$z^\{1-c\}\backslash ;\; \{\}\_2F\_1(1+a-c,1+b-c;2-c;z),$

so the general solution for |z| < 1 is

:$k\backslash ;\; \{\}\_2F\_1(a,b;c;z)+l\; z^\{1-c\}\backslash ;\; \{\}\_2F\_1(1+a-c,1+b-c;2-c;z)$

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,

:$\{\}\_2F\_1(-n,b;c;z)$

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.:

:$\backslash int\_0^x\backslash sqrt\{1+y^\backslash alpha\}\backslash ,\backslash mathrm\{d\}y=\backslash frac\{x\}\{2+\backslash alpha\}\backslash left\; \backslash \{\backslash alpha\backslash ;\{\}\_2F\_1\backslash left(\backslash tfrac\{1\}\{\backslash alpha\},\backslash tfrac\{1\}\{2\};1+\backslash tfrac\{1\}\{\backslash alpha\};-x^\backslash alpha\; \backslash right)\; +2\backslash sqrt\{x^\backslash alpha+1\}\; \backslash right\; \backslash \},\backslash qquad\; \backslash alpha\backslash neq0.$

The Mott polynomials can be written as:See Erdélyi et al. 1955.

:$s\_n(x)=(-x/2)^n\{\}\_3F\_0(-n,\backslash frac\{1-n\}\{2\},1-\backslash frac\{n\}\{2\};;-\backslash frac\{4\}\{x^2\}).$

The function

::$\backslash operatorname\{Li\}\_2(x)\; =\; \backslash sum\_\{n>0\}\backslash ,\{x^n\}\{n^\{-2\}\}\; =\; x\; \backslash ;\; \{\}\_3F\_2(1,1,1;2,2;x)$

is the dilogarithm{{cite web|last=Candan|first=Cagatay|title=A Simple Proof of F(1,1,1;2,2;x)=dilog(1-x)/x

|url=http://www.eee.metu.edu.tr/~ccandan/pub_dir/hyper_rel.pdf}}

The function

::$Q\_n(x;a,b,N)=\; \{\}\_3F\_2(-n,-x,n+a+b+1;a+1,-N+1;1)$

is a Hahn polynomial.

The function

::$p\_n(t^2)=(a+b)\_n(a+c)\_n(a+d)\_n\; \backslash ;\; \{\}\_4F\_3\backslash left(\; -n,\; a+b+c+d+n-1,\; a-t,\; a+t\; ;\; a+b,\; a+c,\; a+d\; ;1\backslash right)$

is a Wilson polynomial.

All roots of a quintic equation can be expressed in terms of radicals and the Bring radical, which is the real solution to $x^5\; +\; x\; +\; a\; =\; 0$. The Bring radical can be written as:

{{cite arXiv

| last = Glasser | first = M. Lawrence

| year = 1994

| title = The quadratic formula made hard: A less radical approach to solving equations

| eprint = math.CA/9411224

}}

::$\backslash operatorname\{BR\}(a)\; =\; -a\; \backslash ;\; \{\}\_4F\_3\backslash left(\; \backslash frac\{1\}\{5\},\; \backslash frac\{2\}\{5\},\; \backslash frac\{3\}\{5\},\; \backslash frac\{4\}\{5\}\; ;\; \backslash frac\{1\}\{2\},\; \backslash frac\{3\}\{4\},\; \backslash frac\{5\}\{4\}\; ;\; \backslash frac\{3125a^4\}\{256\}\; \backslash right).$

The functions

::$\backslash operatorname\{Li\}\_q(z)=z\; \backslash ;\; \{\}\_\{q+1\}F\_q\backslash left(1,1,\backslash ldots,1;2,2,\backslash ldots,2;z\backslash right)$

::$\backslash operatorname\{Li\}\_\{-p\}(z)=z\; \backslash ;\; \{\}\_pF\_\{p-1\}\backslash left(2,2,\backslash ldots,2;1,1,\backslash ldots,1;z\backslash right)$

for $q\backslash in\backslash mathbb\{N\}\_0$ and $p\backslash in\backslash mathbb\{N\}$ are the Polylogarithm.

For each integer n≥2, the roots of the polynomial xⁿ−x+t can be expressed as a sum of at most N−1 hypergeometric functions of type _n+1F_n, which can always be reduced by eliminating at least one pair of a and b parameters.

The generalized hypergeometric function is linked to the Meijer G-function and the MacRobert E-function. Hypergeometric series were generalised to several variables, for example by Paul Emile Appell and Joseph Kampé de Fériet; but a comparable general theory took long to emerge. Many identities were found, some quite remarkable. A generalization, the q-series analogues, called the basic hypergeometric series, were given by Eduard Heine in the late nineteenth century. Here, the ratios considered of successive terms, instead of a rational function of n, are a rational function of qⁿ. Another generalization, the elliptic hypergeometric series, are those series where the ratio of terms is an elliptic function (a doubly periodic meromorphic function) of n.

During the twentieth century this was a fruitful area of combinatorial mathematics, with numerous connections to other fields. There are a number of new definitions of general hypergeometric functions, by Aomoto, Israel Gelfand and others; and applications for example to the combinatorics of arranging a number of hyperplanes in complex N-space (see arrangement of hyperplanes).

Special hypergeometric functions occur as zonal spherical functions on Riemannian symmetric spaces and semi-simple Lie groups. Their importance and role can be understood through the following example: the hypergeometric series ₂F₁ has the Legendre polynomials as a special case, and when considered in the form of spherical harmonics, these polynomials reflect, in a certain sense, the symmetry properties of the two-sphere or, equivalently, the rotations given by the Lie group SO(3). In tensor product decompositions of concrete representations of this group Clebsch–Gordan coefficients are met, which can be written as ₃F₂ hypergeometric series.

Bilateral hypergeometric series are a generalization of hypergeometric functions where one sums over all integers, not just the positive ones.

Fox–Wright functions are a generalization of generalized hypergeometric functions where the Pochhammer symbols in the series expression are generalised to gamma functions of linear expressions in the index n.

• Appell series
• Humbert series
• Kampé de Fériet function
• Lauricella hypergeometric series

{{Reflist|2}}

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| bibcode=2011ZaMP...62...31M| s2cid=30484300| url=https://rke.abertay.ac.uk/en/publications/30e4ad50-271e-40a7-bfb3-dc6515871b50}}

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|last4=Bedford|first4=T. |title=A Bayes linear Bayes Method for Estimation of Correlated Event Rates

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• [https://web.archive.org/web/20060129095451/http://www.cis.upenn.edu/~wilf/AeqB.html The book "A = B"], this book is freely downloadable from the internet.
• MathWorld
• {{MathWorld |title=Generalized Hypergeometric Function |urlname= GeneralizedHypergeometricFunction}}
• {{MathWorld |title=Hypergeometric Function |urlname= HypergeometricFunction}}
• {{MathWorld |title=Confluent Hypergeometric Function of the First Kind |urlname= ConfluentHypergeometricFunctionoftheFirstKind}}
• {{MathWorld |title=Confluent Hypergeometric Limit Function |urlname= ConfluentHypergeometricLimitFunction}}

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