수학 의 특수함수 이론에서 Horn 함수 (Jakob Horn 의 이름)는 Horn (1931) harvtxt 오류 로 열거된 순서 2의 34개의 뚜렷한 수렴초기하계 시리즈 (즉, 두 개의 독립 변수를 가지고 있음)이다. 대상 없음: CITREFHorn1931 (도움말 )CHelprected borngeter (1933 )(Erdelyi 1953 , 섹션 5.7.1) harv error: no target: CITREFerdelyi1953 (도움말 ) B. C. Carlson은[1] [2] 총 34 Horn 함수는 14개의 완전한 초기하 함수와 20개의 합체 초기하 함수로 추가로 분류할 수 있다. 융합영역을 포함하는 전체 기능은 다음과 같다.
F 1 ( α ; β , β ′ ; γ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m + n ( β ) m ( β ′ ) n ( γ ) m + n z m w n m ! n ! / ; z < 1 ∧ w < 1 {\displaystyle F_{1}(\alpha ;\beta ,\beta ';\gamma ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m+n}(\beta )_{m}(\beta ')_{n}}{(\gamma )_{m+n}}}{\frac {z^{m}w^{n}}{m!n! }}/; z <1\land w <1} F 2 ( α ; β , β ′ ; γ , γ ′ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m + n ( β ) m ( β ′ ) n ( γ ) m ( γ ′ ) n z m w n m ! n ! / ; z + w < 1 {\displaystyle F_{2}(\alpha ;\beta ,\beta ';\gamma ,\gamma ';z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m+n}(\beta )_{m}(\beta ')_{n}}{(\gamma )_{m}(\gamma ')_{n}}}{\frac {z^{m}w^{n}}{m!n! }}/; z + w <1} F 3 ( α , α ′ ; β , β ′ ; γ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m ( α ′ ) n ( β ) m ( β ′ ) n ( γ ) m + n z m w n m ! n ! / ; z < 1 ∧ w < 1 {\displaystyle F_{3}(\alpha ,\alpha ';\beta ,\beta ';\gamma ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m}(\alpha ')_{n}(\beta )_{m}(\beta ')_{n}}{(\gamma )_{m+n}}}{\frac {z^{m}w^{n}}{m!n! }}/; z <1\land w <1} F 4 ( α ; β ; γ , γ ′ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m + n ( β ) m + n ( γ ) m ( γ ′ ) n z m w n m ! n ! / ; z + w < 1 {\displaystyle F_{4}(\alpha ;\beta ;\gamma ,\gamma ';z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m+n}(\beta )_{m+n}}{(\gamma )_{m}(\gamma ')_{n}}}{\frac {z^{m}w^{n}}{m!n! }}/{\sqrt{z }+{\sqrt{w}<1} G 1 ( α ; β , β ′ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m + n ( β ) n − m ( β ′ ) m − n z m w n m ! n ! / ; z + w < 1 {\displaystyle G_{1}(\alpha ;\beta ,\beta ';z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }(\alpha )_{m+n}(\beta )_{n-m}(\beta ')_{m-n}{\frac {z^{m}w^{n}}{m!n! }}/; z + w <1} G 2 ( α , α ′ ; β , β ′ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m ( α ′ ) n ( β ) n − m ( β ′ ) m − n z m w n m ! n ! / ; z < 1 ∧ w < 1 {\displaystyle G_{2}(\alpha ,\alpha ';\beta ,\beta ';z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }(\alpha )_{m}(\alpha ')_{n}(\beta )_{n-m}(\beta ')_{m-n}{\frac {z^{m}w^{n}}{m!n! }}/; z <1\land w <1} G 3 ( α , α ′ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 n − m ( α ′ ) 2 m − n z m w n m ! n ! / ; 27 z 2 w 2 + 18 z w ± 4 ( z − w ) < 1 {\displaystyle G_{3}(\alpha ,\alpha ';z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }(\alpha )_{2n-m}(\alpha ')_{2m-n}{\frac {z^{m}w^{n}}{m!n! }}/;27 z ^{2} w ^{2}+18 z w \pm 4(z - w )<1} H 1 ( α ; β ; γ ; δ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( β ) m + n ( γ ) n ( δ ) m z m w n m ! n ! / ; 4 z w + 2 w − w 2 < 1 {\displaystyle H_{1}(\alpha ;\beta ;\gamma ;\delta ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}(\beta )_{m+n}(\gamma )_{n}}{(\delta )_{m}}}{\frac {z^{m}w^{n}}{m!n! }}/;4 z +2 w - w ^{2}<1} H 2 ( α ; β ; γ ; δ ; ϵ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( β ) m ( γ ) n ( δ ) n ( δ ) m z m w n m ! n ! / ; 1 / w − z < 1 {\displaystyle H_{2}(\alpha ;\beta ;\gamma ;\delta ;\epsilon ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}(\beta )_{m}(\gamma )_{n}(\delta )_{n}}{(\delta )_{m}}}{\frac {z^{m}w^{n}}{m!n! }}/;1/ w - z <1} H 3 ( α ; β ; γ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m + n ( β ) n ( γ ) m + n z m w n m ! n ! / ; z + w 2 − w < 0 {\displaystyle H_{3}(\alpha ;\beta ;\gamma ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m+n}(\beta )_{n}}{(\gamma )_{m+n}}}{\frac {z^{m}w^{n}}{m!n! }}/; z + w + w ^{2}- w <0} H 4 ( α ; β ; γ ; δ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m + n ( β ) n ( γ ) m ( δ ) n z m w n m ! n ! / ; 4 z + 2 w − w 2 < 1 {\displaystyle H_{4}(\alpha ;\beta ;\gamma ;\delta ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m+n}(\beta )_{n}}{(\gamma )_{m}(\delta )_{n}}}{\frac {z^{m}w^{n}}{m!n! }}/;4 z +2 w - w ^{2}<1} H 5 ( α ; β ; γ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m + n ( β ) n − m ( γ ) n z m w n m ! n ! / ; 16 z 2 − 36 z w ± ( 8 z − w + 27 z w 2 ) < − 1 {\displaystyle H_{5}(\alpha ;\beta ;\gamma ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m+n}(\beta )_{n-m}}{(\gamma )_{n}}}{\frac {z^{m}w^{n}}{m!n! }}/;16 z ^{2}-36 z w \pm(8 z - w +27 z w w ^{2})<-1} H 6 ( α ; β ; γ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m − n ( β ) n − m ( γ ) n z m w n m ! n ! / ; z w 2 + w < 1 {\displaystyle H_{6}(\alpha ;\beta ;\gamma ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }(\alpha )_{2m-n}(\beta )_{n-m}(\gamma )_{n}{\frac {z^{m}w^{n}}{m!n! }}/; z w ^{2}+ w <1} H 7 ( α ; β ; γ ; δ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m − n ( β ) n ( γ ) n ( δ ) m z m w n m ! n ! / ; 4 z + 2 / s − 1 / s 2 < 1 {\displaystyle H_{7}(\alpha ;\beta ;\gamma ;\delta ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m-n}(\beta )_{n}(\gamma )_{n}}{(\delta )_{m}}}{\frac {z^{m}w^{n}}{m!n! }}/;4 z +2/s -1/s ^{2}<1} 결합 함수는 다음을 포함한다.
Φ 1 ( α ; β ; γ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m + n ( β ) m ( γ ) m + n x m y n m ! n ! {\displaystyle \Phi _{1}\left(\alpha ;\beta ;\gamma ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m+n}(\beta )_{m}}{(\gamma )_{m+n}}}{\frac {x^{m}y^{n}}{m!n! }}} Φ 2 ( β , β ′ ; γ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( β ) m ( β ′ ) n ( γ ) m + n x m y n m ! n ! {\displaystyle \Phi _{2}\left(\beta ,\beta ';\gamma ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\beta )_{m}(\beta ')_{n}}{(\gamma )_{m+n}}}{\frac {x^{m}y^{n}}{m!n! }}} Φ 3 ( β ; γ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( β ) m ( γ ) m + n x m y n m ! n ! {\displaystyle \Phi _{3}\left(\beta ;\gamma ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\beta )_{m}}{(\gamma )_{m+n}}}{\frac {x^{m}y^{n}}{m!n! }}} Ψ 1 ( α ; β ; γ , γ ′ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m + n ( β ) m ( γ ) m ( γ ′ ) n x m y n m ! n ! {\displaystyle \Psi _{1}\left(\alpha ;\beta ;\gamma ,\gamma ';x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m+n}(\beta )_{m}}{(\gamma )_{m}(\gamma ')_{n}}}{\frac {x^{m}y^{n}}{m!n! }}} Ψ 2 ( α ; γ , γ ′ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m + n ( γ ) m ( γ ′ ) n x m y n m ! n ! {\displaystyle \Psi _{2}\left(\alpha ;\gamma ,\gamma ';x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m+n}}{(\gamma )_{m}(\gamma ')_{n}}}{\frac {x^{m}y^{n}}{m!n! }}} Ξ 1 ( α , α ′ ; β ; γ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m ( α ′ ) n ( β ) m ( γ ) m + n ( γ ′ ) n x m y n m ! n ! {\displaystyle \Xi _{1}\left(\alpha ,\alpha ';\beta ;\gamma ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m}(\alpha ')_{n}(\beta )_{m}}{(\gamma )_{m+n}(\gamma ')_{n}}}{\frac {x^{m}y^{n}}{m!n! }}} Ξ 2 ( α ; β ; γ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m ( α ) m ( γ ) m + n x m y n m ! n ! {\displaystyle \Xi _{2}\left(\alpha ;\beta ;\gamma ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m}(\alpha )_{m}}{(\gamma )_{m+n}}}{\frac {x^{m}y^{n}}{m!n! }}} Γ 1 ( α ; β , β ′ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m ( β ) n − m ( β ′ ) m − n x m y n m ! n ! {\displaystyle \Gamma _{1}\left(\alpha ;\beta ,\beta ';x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }(\alpha )_{m}(\beta )_{n-m}(\beta ')_{m-n}{\frac {x^{m}y^{n}}{m!n! }}} Γ 2 ( β , β ′ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( β ) n − m ( β ′ ) m − n x m y n m ! n ! {\displaystyle \Gamma _{2}\left(\beta ,\beta ';x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }(\beta )_{n-m}(\beta ')_{m-n}{\frac {x^{m}y^{n}}{m!n! }}} H 1 ( α ; β ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( β ) m + n ( δ ) m x m y n m ! n ! {\displaystyle H_{1}\left(\alpha ;\beta ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}(\beta )_{m+n}}{(\delta )_{m}}}{\frac {x^{m}y^{n}}{m!n! }}} H 2 ( α ; β ; γ ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( β ) m ( γ ) n ( δ ) m x m y n m ! n ! {\displaystyle H_{2}\left(\alpha ;\beta ;\gamma ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}(\beta )_{m}(\gamma )_{n}}{(\delta )_{m}}}{\frac {x^{m}y^{n}}{m!n! }}} H 3 ( α ; β ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( β ) m ( δ ) m x m y n m ! n ! {\displaystyle H_{3}\left(\alpha ;\beta ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}(\beta )_{m}}{(\delta )_{m}}}{\frac {x^{m}y^{n}}{m!n! }}} H 4 ( α ; γ ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( γ ) n ( δ ) n x m y n m ! n ! {\displaystyle H_{4}\left(\alpha ;\gamma ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}(\gamma )_{n}}{(\delta )_{n}}}{\frac {x^{m}y^{n}}{m!n! }}} H 5 ( α ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( δ ) m x m y n m ! n ! {\displaystyle H_{5}\left(\alpha ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}}{(\delta )_{m}}}{\frac {x^{m}y^{n}}{m!n! }}} H 6 ( α ; γ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m + n ( γ ) m + n x m y n m ! n ! {\displaystyle H_{6}\left(\alpha ;\gamma ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m+n}}{(\gamma )_{m+n}}}{\frac {x^{m}y^{n}}{m!n! }}} H 7 ( α ; γ ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m + n ( γ ) m ( δ ) n x m y n m ! n ! {\displaystyle H_{7}\left(\alpha ;\gamma ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m+n}}{(\gamma )_{m}(\delta )_{n}}}{\frac {x^{m}y^{n}}{m!n! }}} H 8 ( α ; β ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m − n ( β ) n − m x m y n m ! n ! {\displaystyle H_{8}\left(\alpha ;\beta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }(\alpha )_{2m-n}(\beta )_{n-m}{\frac {x^{m}y^{n}}{m!n! }}} H 9 ( α ; β ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m − n ( β ) n ( δ ) m x m y n m ! n ! {\displaystyle H_{9}\left(\alpha ;\beta ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m-n}(\beta )_{n}}{(\delta )_{m}}}{\frac {x^{m}y^{n}}{m!n! }}} H 10 ( α ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m − n ( δ ) m x m y n m ! n ! {\displaystyle H_{10}\left(\alpha ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m-n}}{(\delta )_{m}}}{\frac {x^{m}y^{n}}{m!n! }}} H 11 ( α ; β ; γ ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( β ) n ( γ ) n ( δ ) m x m y n m ! n ! {\displaystyle H_{11}\left(\alpha ;\beta ;\gamma ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}(\beta )_{n}(\gamma )_{n}}{(\delta )_{m}}}{\frac {x^{m}y^{n}}{m!n! }}} 전체 함수와 혼합 함수의 일부는 동일한 표기법을 공유한다는 점에 유의하십시오.
참조 Borngässer, Ludwig (1933), Über hypergeometrische funkionen zweier Veränderlichen , Dissertation, Darmstadt Erdélyi, Arthur; Magnus, Wilhelm ; Oberhettinger, Fritz; Tricomi, Francesco G. (1953), Higher transcendental functions. Vol I (PDF) , McGraw-Hill Book Company, Inc., New York-Toronto-London, MR 0058756 Horn, J. (1935), "Hypergeometrische Funktionen zweier Veränderlichen" , Mathematische Annalen , 105 (1): 381–407, doi :10.1007/BF01455825 J. 혼 수학. Ann. 111 , 637 (1933년)Srivastava, H. M.; Karlsson, Per W. (1985), Multiple Gaussian hypergeometric series , Ellis Horwood Series: Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-85312-602-7 MR 0834385
