阿佩尔函数

**阿佩爾函數**是法國數學家（Paul Apell) 在 1880 年為推廣高斯超幾何函數而創建的一組雙變數函數，定義如下

 

　
: $F_{1}(a,b_{1},b_{2},c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,$

 

　
: $F_{2}(a,b_{1},b_{2},c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m}y^{n}~,$

 

　
: $F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a_{1})_{m}(a_{2})_{n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,$

 

　
: $F_{4}(a,b,c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b)_{m+n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m}y^{n}~.$

 
其中的符號 $:(a)_{m+n}$ 是階乘冪

阿佩爾函數是嫪麗切拉函數和 Kampé\_de\_Fériet 函數的特例。

## 歸遞關係

　
: $(a-b_{1}-b_{2})F_{1}(a,b_{1},b_{2},c;x,y)-a\,F_{1}(a+1,b_{1},b_{2},c;x,y)+b_{1}F_{1}(a,b_{1}+1,b_{2},c;x,y)+b_{2}F_{1}(a,b_{1},b_{2}+1,c;x,y)=0~,$

　
: $c\,F_{1}(a,b_{1},b_{2},c;x,y)-(c-a)F_{1}(a,b_{1},b_{2},c+1;x,y)-a\,F_{1}(a+1,b_{1},b_{2},c+1;x,y)=0~,$

　
: $c\,F_{1}(a,b_{1},b_{2},c;x,y)+c(x-1)F_{1}(a,b_{1}+1,b_{2},c;x,y)-(c-a)x\,F_{1}(a,b_{1}+1,b_{2},c+1;x,y)=0~,$

　
: $c\,F_{1}(a,b_{1},b_{2},c;x,y)+c(y-1)F_{1}(a,b_{1},b_{2}+1,c;x,y)-(c-a)y\,F_{1}(a,b_{1},b_{2}+1,c+1;x,y)=0~.$

其它式子可從這四個關係導出。

 

　
: $c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)+(a_{1}+a_{2}-c)F_{3}(a_{1},a_{2},b_{1},b_{2},c+1;x,y)-a_{1}F_{3}(a_{1}+1,a_{2},b_{1},b_{2},c+1;x,y)-a_{2}F_{3}(a_{1},a_{2}+1,b_{1},b_{2},c+1;x,y)=0~,$

　
: $c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1}+1,a_{2},b_{1},b_{2},c;x,y)+b_{1}x\,F_{3}(a_{1}+1,a_{2},b_{1}+1,b_{2},c+1;x,y)=0~,$

　
: $c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1},a_{2}+1,b_{1},b_{2},c;x,y)+b_{2}y\,F_{3}(a_{1},a_{2}+1,b_{1},b_{2}+1,c+1;x,y)=0~,$

　
: $c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1},a_{2},b_{1}+1,b_{2},c;x,y)+a_{1}x\,F_{3}(a_{1}+1,a_{2},b_{1}+1,b_{2},c+1;x,y)=0~,$

　
: $c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1},a_{2},b_{1},b_{2}+1,c;x,y)+a_{2}y\,F_{3}(a_{1},a_{2}+1,b_{1},b_{2}+1,c+1;x,y)=0~.$

## 導數與微分方程

　
: ${\frac {\partial }{\partial x}}F_{1}(a,b_{1},b_{2},c;x,y)={\frac {ab_{1}}{c}}F_{1}(a+1,b_{1}+1,b_{2},c+1;x,y)~,$

　
: ${\frac {\partial }{\partial y}}F_{1}(a,b_{1},b_{2},c;x,y)={\frac {ab_{2}}{c}}F_{1}(a+1,b_{1},b_{2}+1,c+1;x,y)~.$

 

　
: $\left(x(1-x){\frac {\partial ^{2}}{\partial x^{2}}}+y(1-x){\frac {\partial ^{2}}{\partial x\partial y}}+[c-(a+b_{1}+1)x]{\frac {\partial }{\partial x}}-b_{1}y{\frac {\partial }{\partial y}}-ab_{1}\right)F_{1}(x,y)=0~,$

　
: $\left(y(1-y){\frac {\partial ^{2}}{\partial y^{2}}}+x(1-y){\frac {\partial ^{2}}{\partial x\partial y}}+[c-(a+b_{2}+1)y]{\frac {\partial }{\partial y}}-b_{2}x{\frac {\partial }{\partial x}}-ab_{2}\right)F_{1}(x,y)=0~.$

 

　
: ${\frac {\partial }{\partial x}}F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)={\frac {a_{1}b_{1}}{c}}F_{3}(a_{1}+1,a_{2},b_{1}+1,b_{2},c+1;x,y)~,$

　
: ${\frac {\partial }{\partial y}}F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)={\frac {a_{2}b_{2}}{c}}F_{3}(a_{1},a_{2}+1,b_{1},b_{2}+1,c+1;x,y)~.$

 

　
: $\left(x(1-x){\frac {\partial ^{2}}{\partial x^{2}}}+y{\frac {\partial ^{2}}{\partial x\partial y}}+[c-(a_{1}+b_{1}+1)x]{\frac {\partial }{\partial x}}-a_{1}b_{1}\right)F_{3}(x,y)=0~,$

　
: $\left(y(1-y){\frac {\partial ^{2}}{\partial y^{2}}}+x{\frac {\partial ^{2}}{\partial x\partial y}}+[c-(a_{2}+b_{2}+1)y]{\frac {\partial }{\partial y}}-a_{2}b_{2}\right)F_{3}(x,y)=0~.$

## 積分關係

　
: $F_{1}(a,b_{1},b_{2},c;x,y)={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int _{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b_{1}}(1-yt)^{-b_{2}}\,\mathrm {d} t,\quad \Re \,c>\Re \,a>0~.$

## 特例

　
: $F(\phi ,k)=\int _{0}^{\phi }{\frac {\mathrm {d} \theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}=\sin \phi \,F_{1}({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {3}{2}};\sin ^{2}\phi ,k^{2}\sin ^{2}\phi ),\quad |\Re \,\phi |<{\frac {\pi }{2}}~,$

　
: $E(\phi ,k)=\int _{0}^{\phi }{\sqrt {1-k^{2}\sin ^{2}\theta }}\,\mathrm {d} \theta =\sin \phi \,F_{1}({\tfrac {1}{2}},{\tfrac {1}{2}},-{\tfrac {1}{2}},{\tfrac {3}{2}};\sin ^{2}\phi ,k^{2}\sin ^{2}\phi ),\quad |\Re \,\phi |<{\frac {\pi }{2}}~,$

　
: $\Pi (n,k)=\int _{0}^{\pi /2}{\frac {\mathrm {d} \theta }{(1-n\sin ^{2}\theta ){\sqrt {1-k^{2}\sin ^{2}\theta }}}}={\frac {\pi }{2}}\,F_{1}({\tfrac {1}{2}},1,{\tfrac {1}{2}},1;n,k^{2})~.$

## 參見

Q 阿佩爾函數

## 參考文獻

* Appell, Paul. Sur les séries hypergéométriques de deux variables et sur des équations différentielles linéaires aux dérivées partielles. Comptes rendus hebdomadaires des séances de l'Académie des sciences. 1880, **90**: 296–298 and 731–735. JFM 12.0296.01 （法語）.  (see also "Sur la série F3(α,α',β,β',γ; x,y)" in _C. R. Acad. Sci._ **90**, pp. 977–980)
* Appell, Paul. Sur les fonctions hypergéométriques de deux variables. Journal de Mathématiques Pures et Appliquées. (3ème série). 1882, **8**: 173–216 [2015-04-04]. （原始內容存檔於 2013-04-12） （法語）.
* Appell, Paul; Kampé de Fériet, Joseph. Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite. Paris: Gauthier–Villars. 1926. JFM 52.0361.13 （法語）.  (see p. 14)
* Askey, R. A.; Daalhuis, Adri B. Olde, 阿佩尔函数，Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248
* Bateman, H.; Erdélyi, A. Higher Transcendental Functions, Vol. I (PDF). New York: McGraw–Hill. 1953 [2015-04-04]. （原始內容存檔 (PDF) 於 2011-08-11）.  (see p. 224)
* Gradshteyn, Izrail' Solomonovich; Ryzhik, Iosif Moiseevich. Tablitsy integralov, summ, ryadov i proizvedeniy [Tables of integrals, sums, series and products] 5th. Moscow: Nauka. 1971 （俄語）.  (see Chapter 9.18)
* Humbert, Pierre. Sur les fonctions hypercylindriques. Comptes rendus hebdomadaires des séances de l'Académie des sciences. 1920, **171**: 490–492. JFM 47.0348.01 （法語）.
* Lauricella, Giuseppe. Sulle funzioni ipergeometriche a più variabili. Rendiconti del Circolo Matematico di Palermo. 1893, **7**: 111–158. JFM 25.0756.01. doi:10.1007/BF03012437 （義大利語）.
* Picard, Émile. Sur une extension aux fonctions de deux variables du problème de Riemann relativ aux fonctions hypergéométriques. Annales scientifiques de l'École Normale Supérieure. (2ème série). 1881, **10**: 305–322 [2015-04-04]. JFM 13.0389.01. （原始內容存檔於 2015-01-21） （法語）.  (see also _C. R. Acad. Sci._ **90** (1880), pp. 1119–1121 and 1267–1269)
* Slater, Lucy Joan. Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. 1966. ISBN 0-521-06483-X. MR 0201688.  (there is a 2008 paperback with ISBN 978-0-521-09061-2)