Elliptic analog of hypergeometric series
In mathematics, an elliptic hypergeometric series is a series Σc n such that the ratio c n /c n −1 is an elliptic function of n , analogous to generalized hypergeometric series where the ratio is a rational function of n , and basic hypergeometric series where the ratio is a periodic function of the complex number n . They were introduced by Date-Jimbo-Kuniba-Miwa-Okado (1987) and Frenkel & Turaev (1997) in their study of elliptic 6-j symbols .
For surveys of elliptic hypergeometric series see Gasper & Rahman (2004) , Spiridonov (2008) or Rosengren (2016) .
Definitions [ edit ] The q-Pochhammer symbol is defined by
( 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)(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_{2},\ldots ,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\ldots (a_{m};q)_{n}.} The modified Jacobi theta function with argument x and nome p is defined by
θ ( x ; p ) = ( x , p / x ; p ) ∞ {\displaystyle \displaystyle \theta (x;p)=(x,p/x;p)_{\infty }} 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)} The elliptic shifted factorial is defined by
( 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}} The theta hypergeometric series r +1E r is defined by
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}} The very well poised theta hypergeometric series r +1V r is defined by
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}} The bilateral theta hypergeometric series r G r is defined by
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}} Definitions of additive elliptic hypergeometric series [ edit ] The elliptic numbers are defined by
[ a ; σ , τ ] = θ 1 ( π σ a , e π i τ ) θ 1 ( π σ , e π i τ ) {\displaystyle [a;\sigma ,\tau ]={\frac {\theta _{1}(\pi \sigma a,e^{\pi i\tau })}{\theta _{1}(\pi \sigma ,e^{\pi i\tau })}}} where the Jacobi theta function is defined by
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=-\infty }^{\infty }(-1)^{n}q^{(n+1/2)^{2}}e^{(2n+1)ix}} The additive elliptic shifted factorials are defined by
[ a ; σ , τ ] n = [ a ; σ , τ ] [ a + 1 ; σ , τ ] . . . [ a + n − 1 ; σ , τ ] {\displaystyle [a;\sigma ,\tau ]_{n}=[a;\sigma ,\tau ][a+1;\sigma ,\tau ]...[a+n-1;\sigma ,\tau ]} [ a 1 , . . . , a m ; σ , τ ] = [ a 1 ; σ , τ ] . . . [ a m ; σ , τ ] {\displaystyle [a_{1},...,a_{m};\sigma ,\tau ]=[a_{1};\sigma ,\tau ]...[a_{m};\sigma ,\tau ]} The additive theta hypergeometric series r +1e r is defined by
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}} The additive very well poised theta hypergeometric series r +1v r is defined by
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}} Further reading [ edit ] 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 ]. References [ edit ] 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