초기하계열의 타원형 아날로그
수학에서 타원형 초지하계 시리즈는 비율 n c 가n −1타원 함수 인 경우, 비율이 n 의 합리적 함수 인 일반화된 초지하계 시리즈 와 유사하게, 비율이 복합수 n의 주기 함수인 기본 초지하계 시리즈 와 유사하게 직렬 σc이다. 그것들은 타원형 6-j 기호 에 대한 연구에서 다테-짐보-쿠니바-미와-오카도(1987년) 와 Frenkel & Turaev(1997 )에 의해 소개되었다.
타원형 초지하계 시리즈에 대한 조사는 가스퍼 & 라만(2004 ), 스피리도노프(2008 ) 또는 로젠그렌(2016) 을 참조한다.
정의들 q-Pochhammer 기호 는 다음과 같이 정의된다.
( a ; q ) n = ∏ k = 0 n − 1 ( 1 − a q k ) = ( 1 − a ) ( 1 − a q ) ( 1 − a q 2 ) ⋯ ( 1 − a q n − 1 ) . {\displaystyle \displaystyle (a;q)_{n}=\prod _{k=0}^{n-1}(-1-aq^{k})=(1-a)(1-aq^{2})\cdots(1-aq^{n-1}) } ( a 1 , a 2 , … , a m ; q ) n = ( a 1 ; q ) n ( a 2 ; q ) n … ( a m ; q ) n . {\displaystyle \displaystyle (a_{1},a_{m};q)\ldots ,a_{n}=(a_{1};q)_{n}(a_{n2};q)_{n}\dots(a_{m};q)_{n}}. x 와 nome p 가 있는 수정된 자코비 세타 함수는 다음과 같이 정의된다.
θ ( x ; p ) = ( x , p / x ; p ) ∞ {\displaystyle \displaystyle \theta (x;p)=(x,p/x;p)_{\infit}}} θ ( x 1 , . . . , x m ; p ) = θ ( x 1 ; p ) . . . θ ( x m ; p ) {\displaystyle \displaystyle \theta(x_{1},...,x_{m};p)=\theta(x_{1};p)... \theta (x_{m};p)} 타원 이동 요인 정의:
( a ; q , p ) n = θ ( a ; p ) θ ( a q ; p ) . . . θ ( a q n − 1 ; p ) \displaystyle \displaystyle (a;q,p)_{n}=\theta (a;p)\theta (aq;p)... \theta (aq^{n-1};p)} ( a 1 , . . . , a m ; q , p ) n = ( a 1 ; q , p ) n ⋯ ( a m ; q , p ) n {\displaystyle \displaystyle (a_{1},...,a_{m};q,p)_{n}=(a_{1};q,p)_{n}\cdots(a_{m};q,p)_{n}}}}}}}}}}}}}}}{n}}}}}}}}}}}}}}}}}}} 세타 초기하학 영상 시리즈 E 는r
r + 1 E r ( a 1 , . . . a r + 1 ; b 1 , . . . , b r ; q , p ; z ) = ∑ n = 0 ∞ ( a 1 , . . . , a r + 1 ; q ; p ) n ( q , b 1 , . . . , b r ; q , p ) n z n {\displaystyle \displaystyle {}_{r+1}E_{r}(a_{1},...a_{r+1};b_{1},...,b_{r};q,p;z)=\sum _{n=0}^{\infty }{\frac {(a_{1},...,a_{r+1};q;p)_{n}}{(q,b_{1},...,b_{r};q,p)_{n}}}z^{n}} 매우 침착한 태도의 초기하학 영상 r
r + 1 V r ( a 1 ; a 6 , a 7 , . . . a r + 1 ; q , p ; z ) = ∑ n = 0 ∞ θ ( a 1 q 2 n ; p ) θ ( a 1 ; p ) ( a 1 , a 6 , a 7 , . . . , a r + 1 ; q ; p ) n ( q , a 1 q / a 6 , a 1 q / a 7 , . . . , a 1 q / a r + 1 ; q , p ) n ( q z ) n {\displaystyle \displaystyle {}_{r+1}V_{r}(a_{1};a_{6},a_{7},...a_{r+1};q,p;z)=\sum _{n=0}^{\infty }{\frac {\theta (a_{1}q^{2n};p)}{\theta (a_{1};p)}}{\frac {(a_{1},a_{6},a_{7},...,a_{r+1};q;p)_{n}}{(q,a_{1}q/a_{6},a_{1}q/a_{7},...,a_{1}q/a_{r+1};q,p)_{n}}}(qz)^{n}} 쌍방향 세타 초기하학 영상 r
r G r ( a 1 , . . . a r ; b 1 , . . . , b r ; q , p ; z ) = ∑ n = − ∞ ∞ ( a 1 , . . . , a r ; q ; p ) n ( b 1 , . . . , b r ; q , p ) n z n {\displaystyle \displaystyle {}_{r}G_{r}(a_{1},... a_{r};b_{1},...,b_{r};q,p;z)=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},...,a_{r};q;p)_{n}}{(b_{1},...,b_{r};q,p)_{n}}}z^{n}} 가법 타원형 초지하계 열 정의 타원수는 다음과 같이 정의된다.
[ a ; σ , τ ] = θ 1 ( π σ a , e π i τ ) θ 1 ( π σ , e π i τ ) {\displaystyle [a;\pi ma ,\tau ]={\frac {\fi _{1}(\pi \pi i\tau }}}}{\ta _{1}(\pi \pi \pi \tau }}}}}}}}} 여기서 자코비 세타 함수는
θ 1 ( x , q ) = ∑ n = − ∞ ∞ ( − 1 ) n q ( n + 1 / 2 ) 2 e ( 2 n + 1 ) i x {\displaystyle \theta _{1}(x,q)=\sum _{n=-\nft }^{n=-\nft }^{n1}q^{n+1/2) ^{2}}e^{(2n+1)ix}}} 가법 타원형 이동 요인 정의:
[ a ; σ , τ ] n = [ a ; σ , τ ] [ a + 1 ; σ , τ ] . . . [ a + n − 1 ; σ , τ ] [\displaystyle [a;\buma,\tau ]_{n}=[a;\buma,\tau ][a+1;\buma,\tau ]... [a+n-1;\cHDMA,\tau ]} [ a 1 , . . . , a m ; σ , τ ] = [ a 1 ; σ , τ ] . . . [ a m ; σ , τ ] {\displaystyle [a_{1},...,a_{m};\ma,\tau ]=[a_{1};\ma,\tau ]... [a_{m};\sigma ,\tau ]} 첨가제 Theta 초기하계 영상 시리즈 e 는r
r + 1 e r ( a 1 , . . . a r + 1 ; b 1 , . . . , b r ; σ , τ ; z ) = ∑ n = 0 ∞ [ a 1 , . . . , a r + 1 ; σ ; τ ] n [ 1 , b 1 , . . . , b r ; σ , τ ] n z n {\displaystyle \displaystyle {}_{r+1}e_{r}(a_{1},...a_{r+1};b_{1},...,b_{r};\sigma ,\tau ;z)=\sum _{n=0}^{\infty }{\frac {[a_{1},...,a_{r+1};\sigma ;\tau ]_{n}}{[1,b_{1},...,b_{r};\sigma ,\tau ]_{n}}}z^{n}} 첨가물이 매우 잘 준비되어 있는 초기하학 시리즈 r
r + 1 v r ( a 1 ; a 6 , . . . a r + 1 ; σ , τ ; z ) = ∑ n = 0 ∞ [ a 1 + 2 n ; σ , τ ] [ a 1 ; σ , τ ] [ a 1 , a 6 , . . . , a r + 1 ; σ , τ ] n [ 1 , 1 + a 1 − a 6 , . . . , 1 + a 1 − a r + 1 ; σ , τ ] n z n {\displaystyle \displaystyle {}_{r+1}v_{r}(a_{1};a_{6},...a_{r+1};\sigma ,\tau ;z)=\sum _{n=0}^{\infty }{\frac {[a_{1}+2n;\sigma ,\tau ]}{[a_{1};\sigma ,\tau ]}}{\frac {[a_{1},a_{6},...,a_{r+1};\sigma ,\tau ]_{n}}{[1,1+a_{1}-a_{6},...,1+a_{1}-a_{r+1};\sigma ,\tau ]_{n}}}z^{n}} 추가 읽기 Spiridonov, V. P. (2013). "Aspects of elliptic hypergeometric functions". In Berndt, Bruce C. (ed.). The Legacy of Srinivasa Ramanujan Proceedings of an International Conference in Celebration of the 125th Anniversary of Ramanujan's Birth ; University of Delhi, 17-22 December 2012 . Ramanujan Mathematical Society Lecture Notes Series. Vol. 20. Ramanujan Mathematical Society. pp. 347–361. arXiv :1307.2876 Bibcode :2013arXiv1307.2876S . ISBN 9789380416137 Rosengren, Hjalmar (2016). "Elliptic Hypergeometric Functions". arXiv :1608.06161 math.CA ]. 참조 Frenkel, Igor B.; Turaev, Vladimir G. (1997), "Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions" , The Arnold-Gelfand mathematical seminars , Boston, MA: Birkhäuser Boston, pp. 171–204, ISBN 978-0-8176-3883-2 MR 1429892 Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series , Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press , ISBN 978-0-521-83357-8 MR 2128719 Spiridonov, V. P. (2002), "Theta hypergeometric series", Asymptotic combinatorics with application to mathematical physics (St. Petersburg, 2001) , NATO Sci. Ser. II Math. Phys. Chem., vol. 77, Dordrecht: Kluwer Acad. Publ., pp. 307–327, arXiv :math/0303204 Bibcode :2003math......3204S , MR 2000728 Spiridonov, V. P. (2003), "Theta hypergeometric integrals", Rossiĭskaya Akademiya Nauk. Algebra i Analiz , 15 (6): 161–215, arXiv :math/0303205 Bibcode :2003math......3205S , doi :10.1090/S1061-0022-04-00839-8 , MR 2044635 , S2CID 14471695 Spiridonov, V. P. (2008), "Essays on the theory of elliptic hypergeometric functions", Rossiĭskaya Akademiya Nauk. Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk , 63 (3): 3–72, arXiv :0805.3135 Bibcode :2008RuMaS..63..405S , doi :10.1070/RM2008v063n03ABEH004533 , MR 2479997 , S2CID 16996893 Warnaar, S. Ole (2002), "Summation and transformation formulas for elliptic hypergeometric series", Constructive Approximation , 18 (4): 479–502, arXiv :math/0001006 doi :10.1007/s00365-002-0501-6 , MR 1920282 , S2CID 18102177
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