阿佩尔函数

阿佩尔函数是法国数学家（Paul Apell)在1880年为推广高斯超几何函数而创建的一组雙变数函数，定义如下 F 1 ( a , b 1 , b 2 , c ; x , y ) = ∑ m , n = 0 ∞ ( a ) m + n ( b 1 ) m ( b 2 ) n ( c ) m + n m ! n ! x m y n   , {\displaystyle 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 ) = ∑ m , n = 0 ∞ ( a ) m + n ( b 1 ) m ( b 2 ) n ( c 1 ) m ( c 2 ) n m ! n ! x m y n   , {\displaystyle 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 ) = ∑ m , n = 0 ∞ ( a 1 ) m ( a 2 ) n ( b 1 ) m ( b 2 ) n ( c ) m + n m ! n ! x m y n   , {\displaystyle 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 ) = ∑ m , n = 0 ∞ ( a ) m + n ( b ) m + n ( c 1 ) m ( c 2 ) n m ! n ! x m y n   . {\displaystyle 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}~.} 其中的符号 : ( a ) m + n {\displaystyle :(a)_{m+n}} 是阶乘幂 阿佩尔函数是嫪丽切拉函数和Kampé_de_Fériet函数的特例。 ( 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   , {\displaystyle (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   , {\displaystyle 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   , {\displaystyle 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   . {\displaystyle 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   , {\displaystyle 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   , {\displaystyle 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   , {\displaystyle 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   , {\displaystyle 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   . {\displaystyle 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~.} ∂ ∂ x F 1 ( a , b 1 , b 2 , c ; x , y ) = a b 1 c F 1 ( a + 1 , b 1 + 1 , b 2 , c + 1 ; x , y )   , {\displaystyle {\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)~,} ∂ ∂ y F 1 ( a , b 1 , b 2 , c ; x , y ) = a b 2 c F 1 ( a + 1 , b 1 , b 2 + 1 , c + 1 ; x , y )   . {\displaystyle {\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)~.} ( x ( 1 − x ) ∂ 2 ∂ x 2 + y ( 1 − x ) ∂ 2 ∂ x ∂ y + [ c − ( a + b 1 + 1 ) x ] ∂ ∂ x − b 1 y ∂ ∂ y − a b 1 ) F 1 ( x , y ) = 0   , {\displaystyle \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~,} ( y ( 1 − y ) ∂ 2 ∂ y 2 + x ( 1 − y ) ∂ 2 ∂ x ∂ y + [ c − ( a + b 2 + 1 ) y ] ∂ ∂ y − b 2 x ∂ ∂ x − a b 2 ) F 1 ( x , y ) = 0   . {\displaystyle \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~.} ∂ ∂ x F 3 ( a 1 , a 2 , b 1 , b 2 , c ; x , y ) = a 1 b 1 c F 3 ( a 1 + 1 , a 2 , b 1 + 1 , b 2 , c + 1 ; x , y )   , {\displaystyle {\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)~,} ∂ ∂ y F 3 ( a 1 , a 2 , b 1 , b 2 , c ; x , y ) = a 2 b 2 c F 3 ( a 1 , a 2 + 1 , b 1 , b 2 + 1 , c + 1 ; x , y )   . {\displaystyle {\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)~.} ( x ( 1 − x ) ∂ 2 ∂ x 2 + y ∂ 2 ∂ x ∂ y + [ c − ( a 1 + b 1 + 1 ) x ] ∂ ∂ x − a 1 b 1 ) F 3 ( x , y ) = 0   , {\displaystyle \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~,} ( y ( 1 − y ) ∂ 2 ∂ y 2 + x ∂ 2 ∂ x ∂ y + [ c − ( a 2 + b 2 + 1 ) y ] ∂ ∂ y − a 2 b 2 ) F 3 ( x , y ) = 0   . {\displaystyle \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~.} F 1 ( a , b 1 , b 2 , c ; x , y ) = Γ ( c ) Γ ( a ) Γ ( c − a ) ∫ 0 1 t a − 1 ( 1 − t ) c − a − 1 ( 1 − x t ) − b 1 ( 1 − y t ) − b 2 d t , ℜ c > ℜ a > 0   . {\displaystyle 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~.} F ( ϕ , k ) = ∫ 0 ϕ d θ 1 − k 2 sin 2 ⁡ θ = sin ⁡ ϕ F 1 ( 1 2 , 1 2 , 1 2 , 3 2 ; sin 2 ⁡ ϕ , k 2 sin 2 ⁡ ϕ ) , | ℜ ϕ | < π 2   , {\displaystyle 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 ( ϕ , k ) = ∫ 0 ϕ 1 − k 2 sin 2 ⁡ θ d θ = sin ⁡ ϕ F 1 ( 1 2 , 1 2 , − 1 2 , 3 2 ; sin 2 ⁡ ϕ , k 2 sin 2 ⁡ ϕ ) , | ℜ ϕ | < π 2   , {\displaystyle 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}}~,} Π ( n , k ) = ∫ 0 π / 2 d θ ( 1 − n sin 2 ⁡ θ ) 1 − k 2 sin 2 ⁡ θ = π 2 F 1 ( 1 2 , 1 , 1 2 , 1 ; n , k 2 )   . {\displaystyle \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})~.} Q阿佩尔函数 例如： ( y − x ) F 1 ( a , b 1 + 1 , b 2 + 1 , c , x , y ) = y F 1 ( a , b 1 , b 2 + 1 , c , x , y ) − x F 1 ( a , b 1 + 1 , b 2 , c , x , y ) {\displaystyle (y-x)F_{1}(a,b_{1}+1,b_{2}+1,c,x,y)=y\,F_{1}(a,b_{1},b_{2}+1,c,x,y)-x\,F_{1}(a,b_{1}+1,b_{2},c,x,y)} 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 （法语）.  (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]. （原始内容存档于2013-04-12） （法语）.  Appell, Paul; Kampé de Fériet, Joseph. Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite. Paris: Gauthier–Villars. 1926. JFM 52.0361.13 （法语）.  (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. (编), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248  Bateman, H.; Erdélyi, A. Higher Transcendental Functions, Vol. I (PDF). 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