Generalized hypergeometric function: Difference between revisions

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{{Short description|Family of power series in mathematics}}

{{For|other generalizations of the hypergeometric function|hypergeometric function}}

{{Distinguish|general hypergeometric function}}{{Redirects here|PFq|3=PFQ (disambiguation)}}[[File:Plot of the generalized hypergeometric function pFq(a b z) with a=(2,4,6,8) and b=(2,3,5,7,11) in the complex plane from -2-2i to 2+2i created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the generalized hypergeometric function pFq(a b z) with a=(2,4,6,8) and b=(2,3,5,7,11) in the complex plane from -2-2i to 2+2iPlot of the generalized hypergeometric function pFq(a b z) with a=(2,4,6,8) and b=(2,3,5,7,11) in the complex plane from -2-2i to 2+2i created with Mathematica 13.1 function ComplexPlot3D created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the generalized hypergeometric function pFq(a b z) with a=(2,4,6,8) and b=(2,3,5,7,11) in the complex plane from -2-2i to 2+2i created with Mathematica 13.1 function ComplexPlot3D]]

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:<math>\begin{align}

(a)_0 &= 1, \\

(a)_n &= a(a+1)(a+2) \cdots (a+n-1) = \frac{\Gamma(a+n)}{\Gamma(a)}, && n \geq 1,

\end{align}</math>

where <math>\Gamma(x)</math> represents the [[Gamma function|gamma function]], this can be written

this can be written

:<math>\,{}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\cdots(a_p)_n}{(b_1)_n\cdots(b_q)_n} \, \frac {z^n} {n!} = \frac{\Gamma(b_1)\cdots \Gamma(b_q)}{\Gamma(a_1)\cdots \Gamma(a_p)} \sum_{n=0}^{\infty} \frac{\Gamma(n+a_1)\cdots \Gamma(n+a_p)}{\Gamma(n+b_1)\cdots \Gamma(n+b_q)} \frac{z^n}{n!}.</math>

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

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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'' &lt; ''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.

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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,

:<math>\,{}_2F_1(3,1;1;z) = \,{}_2F_1(1,3;1;z) = \,{}_1F_0(3;;z)</math>.

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.<ref>{{cite book|firstfirst1=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}}</ref><ref>{{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 }}</ref>

:<math>

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

Line 83 ⟶ 84:

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

\end{array}

;z\right] = \sum_{j = 0}^n \binom{n}{j} \frac{1z^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 \\

Line 124 ⟶ 125:

\end{align}</math>

Combining these gives a differential equation satisfied by ''w'' = _''p''''F''_''q'':

:<math>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</math>.

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:<math>z\; {}_pF_q(a_1+1,\dots,a_p+1;b_1+1,\dots,b_q+1;z),</math>

:<math>{}_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z).</math>

Since the space has dimension 2, any three of these ''p''+''q''+2 functions are linearly dependent. These dependencies can be written out to generate a large number of identities involving <math>{}_pF_q</math>.:

<ref>{{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 }}</ref><ref>

{{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}}</ref>

:<math>

(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).

</math>

:<math>

(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).

</math>

:<math>

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)

.

</math>

:<math>

(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)

.

</math>

These dependencies can be written out to generate a large number of identities involving <math>{}_pF_q</math>.

For example, in the simplest non-trivial case,

Line 165 ⟶ 201:

is called '''contiguous''' to

:<math>{}_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z).</math>

Using the technique outlined above, an identity relating <math>{}_0F_1(;a;z)</math> and its two contiguous functions can be given, six identities relating <math>{}_1F_1(a;b;z)</math> and any two of its four contiguous functions, and fifteen identities relating <math>{}_2F_1(a,b;c;z)</math> 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 in his 1812 paper|1813}}.)

==Identities==

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===Saalschütz's theorem===

Saalschütz's theorem<ref>See {{harv|Slater|1966|loc=Section&nbsp;2.3.1}} or {{harv|Bailey|1935|loc=Section&nbsp;2.2}} for a proof, or the [[proofwiki:Pfaff-Saalschütz_Theorem|ProofWiki]].</ref> {{harv|Saalschütz|1890}} is

:<math>{}_3F_2 (a,b, -n;c, 1+a+b-c-n;1)= \frac{(c-a)_n(c-b)_n}{(c)_n(c-a-b)_n}.</math>

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 [[Johann Friedrich Pfaff|Pfaff]] in 1797.<ref>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.) </ref>

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

===Dixon's identity===

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

:<math>1+2a=b+c+d+e-m.</math>

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.<ref>{{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}}</ref>

===Generalization of Kummer's transformations and identities for ₂''F''₂===

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When a is a non-positive integer, −''n'', <math>{}_1F_1(-n;b;z)</math> is a polynomial. Up to constant factors, these are the [[Laguerre polynomials]]. This implies [[Hermite polynomials]] can be expressed in terms of ₁''F''₁ as well.

===The series ₁''F''₂===

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

The function <math>x\; {}_1F_2\left(\frac{1}{2};\frac{3}{2},\frac{3}{2};-\frac{x^2}{4}\right)</math> is the antiderivative of the [[cardinal sine]]. With modified values of <math>a_1</math> and <math>b_1</math>, one obtains the antiderivative of <math>\sin(x^\beta)/x^\alpha</math>.<ref>Victor Nijimbere, Ural Math J vol 3(1) and https://arxiv.org/abs/1703.01907 (2017)</ref>

The [[Lommel function]] is <math> s_{\mu, \nu} (z) = \frac{z^{\mu + 1}}{(\mu - \nu + 1)(\mu + \nu + 1)} {}_1F_2(1; \frac{\mu}{2} - \frac{\nu}{2} + \frac{3}{2} , \frac{\mu}{2} + \frac{\nu}{2} + \frac{3}{2} ;-\frac{z^2}{4}) </math>.<ref>Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)</ref>

===The series ₂''F''₀===

The confluent hypergeometric function of the second kind can be written as:<ref>{{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}}</ref>

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

:<math>U(a,b,z) = z^{-a} \; {}_2 F_0 \left( a, a-b+1; ; -\frac{1}{z}\right).</math>

===The series ₂''F''₁===

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:<math>\int_0^x\sqrt{1+y^\alpha}\,\mathrm{d}y=\frac{x}{2+\alpha}\left \{\alpha\;{}_2F_1\left(\tfrac{1}{\alpha},\tfrac{1}{2};1+\tfrac{1}{\alpha};-x^\alpha \right) +2\sqrt{x^\alpha+1} \right \},\qquad \alpha\neq0.</math>

===The series ₂''F''₂===

The hypergeometric series <math>{}_2F_2</math> is generally associated with integrals of products of power functions and the exponential function. As such, the [[Exponential integral|exponential integral]] can be written as:

:<math>\operatorname{Ei}(x)=x{}_2F_2(1,1;2,2;x)+\ln x+\gamma.</math>

===The series ₃''F''₀===

This occurs in connection withThe [[Mott polynomials]]. can be written as:<ref>See Erdélyi et al. 1955.</ref>

:<math>s_n(x)=(-x/2)^n{}_3F_0(-n,\frac{1-n}{2},1-\frac{n}{2};;-\frac{4}{x^2}).</math>

===The series ₃''F''₁===

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

===The series ₃''F''₂===

The function

::<math>\operatorname{Li}_2(x) = \sum_{n>0}\,{x^n}{n^{-2}} = x \; {}_3F_2(1,1,1;2,2;x)</math> is the [[dilogarithm]]<ref>{{cite web|last=Candan|first=Cagatay|title=A Simple Proof of F(1,1,1;2,2;x)=dilog(1-x)/x

::<math>\operatorname{Li}_2(x) = \sum_{n>0}\,{x^n}{n^{-2}} = x \; {}_3F_2(1,1,1;2,2;x)</math>

is the [[dilogarithm]]<ref>{{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}}</ref>

The function

::<math>Q_n(x;a,b,N)= {}_3F_2(-n,-x,n+a+b+1;a+1,-N+1;1)</math> is a [[Hahn polynomial]].

::<math>Q_n(x;a,b,N)= {}_3F_2(-n,-x,n+a+b+1;a+1,-N+1;1)</math>

is a [[Hahn polynomial]].

===The series ₄''F''₃===

The function

::<math>p_n(t^2)=(a+b)_n(a+c)_n(a+d)_n \; {}_4F_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)</math> is a [[Wilson polynomial]].

::<math>p_n(t^2)=(a+b)_n(a+c)_n(a+d)_n \; {}_4F_3\left( -n, a+b+c+d+n-1, a-t, a+t ; a+b, a+c, a+d ;1\right)</math>

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 <math>x^5 + x + a = 0</math>. The Bring radical can be written as:<ref name="BR">

{{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

}}

</ref>

::<math>\operatorname{BR}(a) = -a \; {}_4F_3\left( \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} ; \frac{1}{2}, \frac{3}{4}, \frac{5}{4} ; \frac{3125a^4}{256} \right).</math>

The partition function <math>Z(K)</math> of the 2D isotropic [[Square lattice Ising model|Ising model]] with no external magnetic field was found by [[Lars Onsager|Onsager]] in the 1940s and can be expressed as<ref name="Ising">

{{cite journal

| last = Viswanathan | first = G. M.

| year = 2014

| title = The hypergeometric series for the partition function of the 2-D Ising model

| journal = Journal of Statistical Mechanics: Theory and Experiment

| volume = 2015

| issue = 7

| page = 07004

| doi = 10.1088/1742-5468/2015/07/P07004

| arxiv = 1411.2495

| bibcode = 2015JSMTE..07..004V

}}

</ref>

::<math>\ln Z(K) = \ln(2 \cosh 2K) -k^2 {}_4F_3\left( 1, 1, \frac{3}{2}, \frac{3}{2} ; 2, 2, 2; 16k^2 \right),</math>

with <math>K=\frac{J}{k_\mathrm{B}T}</math> and <math>k=\frac{1}{2}\tanh 2K\,\operatorname{sech} 2K</math>.

===The series _q+1''F''_q===

The functions

::<math>\operatorname{Li}_q(z)=z \; {}_{q+1}F_q\left(1,1,\ldots,1;2,2,\ldots,2;z\right)</math>

::<math>\operatorname{Li}_{-p}(z)=z \; {}_pF_{p-1}\left(2,2,\ldots,2;1,1,\ldots,1;z\right)</math>

for <math>q\in\mathbb{N}_0</math> and <math>p\in\mathbb{N}</math> are the [[Polylogarithm]].

For each integer ''n''≥2, the roots of the polynomial ''x''^''n''−''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.<ref name="BR"/>

==Generalizations==

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*{{Cite book | last1=Erdélyi | first1=Arthur | last2=Magnus | first2=Wilhelm | author2-link=Wilhelm Magnus | last3=Oberhettinger | first3=Fritz | last4=Tricomi | first4=Francesco G. | title=Higher transcendental functions. Vol. III | publisher=McGraw-Hill Book Company, Inc., New York-Toronto-London | mr=0066496 | year=1955}}

* {{cite book | last1= Gasper | first1= George | last2= Rahman | first2= Mizan | author-link2= Mizan Rahman | title= Basic Hypergeometric Series | edition= 2nd | year= 2004 | series= Encyclopedia of Mathematics and Its Applications | volume= 96 | publisher= Cambridge University Press | ___location= Cambridge, UK | mr= 2128719 | zbl= 1129.33005 | isbn= 978-0-521-83357-8 }} (the first edition has {{isbn|0-521-35049-2}})

* {{cite journal | last= Gauss | first= Carl Friedrich | author-link= Carl Friedrich Gauss | title= Disquisitiones generales circa seriam infinitam &nbsp; <math> 1 + \tfrac {\alpha \beta} {1 \cdot \gamma} ~x + \tfrac {\alpha (\alpha+1) \beta (\beta+1)} {1 \cdot 2 \cdot \gamma (\gamma+1)} ~x~x + \mbox{etc.} </math> | language= la | url= https://books.google.com/books?id=uDMAAAAAQAAJ | journal= Commentationes Societatis Regiae Scientarum Gottingensis Recentiores | ___location= Göttingen | year= 1813 | volume= 2 }} (a reprint of this paper can be found in [https://archive.org/details/bub_gb_uDMAAAAAQAAJ ''Carl Friedrich Gauss, Werke''], p.&nbsp;125) (a translation is available [[wikisource:Translation:Disquisitiones_generales_circa_seriem_infinitam_...|on Wikisource]])

*{{Citation | last1=Grinshpan | first1=A. Z. | title=Generalized hypergeometric functions: product identities and weighted norm inequalities | doi=10.1007/s11139-013-9487-x | year=2013 |

journal=The Ramanujan Journal | volume=31 | issue=1–2 | pages=53–66 | s2cid=121054930 }}

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* {{cite journal|last1=Quigley |first1=J.|last2=Wilson |first2=K.J. |last3=Walls |first3=L.

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

|journal=Risk Analysis |volume=33|issue=12|pages=2209–2224|year=2013|doi=10.1111/risa.12035|pmid=23551053|bibcode=2013RiskA..33.2209Q |s2cid=24476762 |url=https://strathprints.strath.ac.uk/43403/1/EventRatesRA.pdf}}

* {{cite journal | last1= Rathie | first1= Arjun K. | last2= Pogány | first2= Tibor K. | title= New summation formula for ₃''F''₂(1/2) and a Kummer-type II transformation of ₂''F''₂(''x'') | journal = Mathematical Communications | volume= 13 | year= 2008 | pages= 63–66 | url= http://hrcak.srce.hr/file/37118 | mr= 2422088 | zbl= 1146.33002 }}

* {{cite journal|last1=Rakha |first1=M.A.|last2=Rathie |first2=Arjun K.|title=Extensions of Euler's type- II transformation and Saalschutz's theorem |journal=Bull. Korean Math. Soc.|volume=48 |number=1|pages=151–156|year=2011 |doi=10.4134/bkms.2011.48.1.151|doi-access=free}}

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[[Category:Hypergeometric functions|*]]

[[Category:Ordinary differential equations]]

[[Category:MathematicalSeries series(mathematics)]]