Humbert series

In mathematics, '''Humbert series''' are a set of seven hypergeometric series Φ₁, Φ₂, Φ₃, Ψ₁, Ψ₂, Ξ₁, Ξ₂ of two variables that generalize Kummer's confluent hypergeometric series ₁''F''₁ of one variable and the confluent hypergeometric limit function ₀''F''₁ of one variable. The first of these double series was introduced by {{harvs|txt|authorlink=Pierre Humbert (mathematician)|first=Pierre|last= Humbert|year=1920}}.

==Definitions== The Humbert series Φ₁ is defined for |''x''| < 1 by the double series:

:<math> \Phi_1(a,b,c;x,y) = F_1(a,b,-,c;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b)_m} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~, </math>

where the Pochhammer symbol (''q'')_''n'' represents the rising factorial:

:<math>(q)_n = q\,(q+1) \cdots (q+n-1) = \frac{\Gamma(q+n)}{\Gamma(q)}~,</math>

where the second equality is true for all complex <math>q</math> except <math>q=0,-1,-2,\ldots</math>. For other values of ''x'' the function Φ₁ can be defined by analytic continuation.

The Humbert series Φ₁ can also be written as a one-dimensional Euler-type integral:

:<math> \Phi_1(a,b,c;x,y) = \frac{\Gamma(c)} {\Gamma(a) \Gamma(c-a)} \int_0^1 t^{a-1} (1-t)^{c-a-1} (1-xt)^{-b} e^{yt} \,\mathrm{d}t, \quad \real \,c > \real \,a > 0 ~. </math>

This representation can be verified by means of Taylor expansion of the integrand, followed by termwise integration.

Similarly, the function Φ₂ is defined for all ''x'', ''y'' by the series:

:<math> \Phi_2(b_1,b_2,c;x,y) = F_1(-,b_1,b_2,c;x,y) = \sum_{m,n=0}^\infty \frac{(b_1)_m (b_2)_n} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~, </math> the function Φ₃ for all ''x'', ''y'' by the series:

:<math> \Phi_3(b,c;x,y) = \Phi_2(b,-,c;x,y) = F_1(-,b,-,c;x,y) = \sum_{m,n=0}^\infty \frac{(b)_m} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~, </math>

the function Ψ₁ for |''x''| < 1 by the series:

:<math> \Psi_1(a,b,c_1,c_2;x,y) = F_2(a,b,-,c_1,c_2;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b)_m} {(c_1)_m (c_2)_n \,m! \,n!} \,x^m y^n ~, </math>

the function Ψ₂ for all ''x'', ''y'' by the series:

:<math> \Psi_2(a,c_1,c_2;x,y) = \Psi_1(a,-,c_1,c_2;x,y) = F_2(a,-,-,c_1,c_2;x,y) = F_4(a,-,c_1,c_2;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n}} {(c_1)_m (c_2)_n \,m! \,n!} \,x^m y^n ~, </math>

the function Ξ₁ for |''x''| < 1 by the series:

:<math> \Xi_1(a_1,a_2,b,c;x,y) = F_3(a_1,a_2,b,-,c;x,y) = \sum_{m,n=0}^\infty \frac{(a_1)_m (a_2)_n (b)_m} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~, </math>

and the function Ξ₂ for |''x''| < 1 by the series:

:<math> \Xi_2(a,b,c;x,y) = \Xi_1(a,-,b,c;x,y) = F_3(a,-,b,-,c;x,y) = \sum_{m,n=0}^\infty \frac{(a)_m (b)_m} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~. </math>

==Related series== * {{main|Appell series}} :There are four related series of two variables, ''F''₁, ''F''₂, ''F''₃, and ''F''₄, which generalize Gauss's hypergeometric series ₂''F''₁ of one variable in a similar manner and which were introduced by Paul Émile Appell in 1880.

==References== * {{cite book | last1= Appell | first1= Paul | author1-link= Paul Émile Appell | last2= Kampé de Fériet | first2= Joseph | author2-link= Joseph Kampé de Fériet | title= Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite | language= French | location= Paris | publisher= Gauthier–Villars | year= 1926 | jfm= 52.0361.13 }} (see p.&nbsp;126) * {{cite book | first1= H. | last1= Bateman | author1-link= Harry Bateman | first2= A. | last2= Erdélyi | author2-link= Arthur Erdélyi | title= Higher Transcendental Functions, Vol. I | url= http://apps.nrbook.com/bateman/Vol1.pdf | location= New York | publisher= McGraw–Hill | year= 1953 | access-date= 2012-05-23 | archive-date= 2011-08-11 | archive-url= https://web.archive.org/web/20110811153220/http://apps.nrbook.com/bateman/Vol1.pdf | url-status= dead }} (see p.&nbsp;225) *{{cite book |author-first1=Izrail Solomonovich |author-last1=Gradshteyn |author-link1=Izrail Solomonovich Gradshteyn |author-first2=Iosif Moiseevich |author-last2=Ryzhik |author-link2=Iosif Moiseevich Ryzhik |author-first3=Yuri Veniaminovich |author-last3=Geronimus |author-link3=Yuri Veniaminovich Geronimus |author-first4=Michail Yulyevich |author-last4=Tseytlin |author-link4=Michail Yulyevich Tseytlin |author-first5=Alan |author-last5=Jeffrey |editor-first1=Daniel |editor-last1=Zwillinger |editor-first2=Victor Hugo |editor-last2=Moll |editor-link2=Victor Hugo Moll |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=Academic Press, Inc. |date=2015 |orig-year=October 2014 |edition=8 |language=English |isbn=978-0-12-384933-5 |lccn=2014010276 <!-- |url=https://books.google.com/books?id=NjnLAwAAQBAJ |access-date=2016-02-21-->|title-link=Gradshteyn and Ryzhik |chapter=9.26. }} * {{cite journal | last= Humbert | first= Pierre | author-link= Pierre Humbert (mathematician) | title= Sur les fonctions hypercylindriques | language= French | journal= Comptes rendus hebdomadaires des séances de l'Académie des sciences | year= 1920 | volume= 171 | pages= 490&ndash;492 | jfm= 47.0348.01 }}

{{DEFAULTSORT:Humbert Series}} Category:Hypergeometric functions Category:Series (mathematics)