Elliptic 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 Frenkel & Turaev (1997) in their study of elliptic 6-j symbols.

For surveys of elliptic hypergeometric series see Gasper & Rahman (2004) or Spiridonov (2008).

Definitions

The q-Pochhammer symbol is defined by

$\displaystyle (a;q)_{n}=\prod _{{k=0}}^{{n-1}}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\cdots (1-aq^{{n-1}}).$

$\displaystyle (a_{1},a_{2},\ldots ,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\ldots (a_{m};q)_{n}.$

The modified Jacobi theta function with argument x and nome p is defined by

$\displaystyle \theta (x;p)=(x,p/x;p)_{\infty }$

$\displaystyle \theta (x_{1},...,x_{m};p)=\theta (x_{1};p)...\theta (x_{m};p)$

The elliptic shifted factorial is defined by

$\displaystyle (a;q,p)_{n}=\theta (a;p)\theta (aq;p)...\theta (aq^{{n-1}};p)$

$\displaystyle (a_{1},...,a_{m};q,p)_{n}=(a_{1};q,p)_{n}\cdots (a_{m};q,p)_{n}$

The theta hypergeometric series _r+1E_r is defined by

$\displaystyle {}_{{r+1}}E_{r}(a_{1},...a_{{r+1}};b_{1},...,b_{r};q,p;z)=\sum _{{n=0}}^{\infty }{\frac {(a_{1},...,a_{{r+1}};q;p)_{n}}{(q,b_{1},...,b_{r};q,p)_{n}}}z^{n}$

The very well poised theta hypergeometric series _r+1V_r is defined by

$\displaystyle {}_{{r+1}}V_{r}(a_{1};a_{6},a_{7},...a_{{r+1}};q,p;z)=\sum _{{n=0}}^{\infty }{\frac {\theta (a_{1}q^{{2n}};p)}{\theta (a_{1};p)}}{\frac {(a_{1},a_{6},a_{7},...,a_{{r+1}};q;p)_{n}}{(q,a_{1}q/a_{6},a_{1}q/a_{7},...,a_{1}q/a_{{r+1}};q,p)_{n}}}(qz)^{n}$

The bilateral theta hypergeometric series _rG_r is defined by

$\displaystyle {}_{{r}}G_{r}(a_{1},...a_{{r}};b_{1},...,b_{r};q,p;z)=\sum _{{n=-\infty }}^{\infty }{\frac {(a_{1},...,a_{{r}};q;p)_{n}}{(b_{1},...,b_{r};q,p)_{n}}}z^{n}$

Definitions of additive elliptic hypergeometric series

The elliptic numbers are defined by

$[a;\sigma ,\tau ]={\frac {\theta _{1}(\pi \sigma a,e^{{\pi i\tau }})}{\theta _{1}(\pi \sigma ,e^{{\pi i\tau }})}}$

where the Jacobi theta function is defined by

$\theta _{1}(x,q)=\sum _{{n=-\infty }}^{\infty }(-1)^{n}q^{{(n+1/2)^{2}}}e^{{(2n+1)ix}}$

The additive elliptic shifted factorials are defined by

• $[a;\sigma ,\tau ]_{n}=[a;\sigma ,\tau ][a+1;\sigma ,\tau ]...[a+n-1;\sigma ,\tau ]$
• $[a_{1},...,a_{m};\sigma ,\tau ]=[a_{1};\sigma ,\tau ]...[a_{m};\sigma ,\tau ]$

The additive theta hypergeometric series _r+1e_r is defined by

$\displaystyle {}_{{r+1}}e_{r}(a_{1},...a_{{r+1}};b_{1},...,b_{r};\sigma ,\tau ;z)=\sum _{{n=0}}^{\infty }{\frac {[a_{1},...,a_{{r+1}};\sigma ;\tau ]_{n}}{[1,b_{1},...,b_{r};\sigma ,\tau ]_{n}}}z^{n}$

The additive very well poised theta hypergeometric series _r+1v_r is defined by

$\displaystyle {}_{{r+1}}v_{r}(a_{1};a_{6},...a_{{r+1}};\sigma ,\tau ;z)=\sum _{{n=0}}^{\infty }{\frac {[a_{1}+2n;\sigma ,\tau ]}{[a_{1};\sigma ,\tau ]}}{\frac {[a_{1},a_{6},...,a_{{r+1}};\sigma ,\tau ]_{n}}{[1,1+a_{1}-a_{6},...,1+a_{1}-a_{{r+1}};\sigma ,\tau ]_{n}}}z^{n}$

References

• 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, 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., 77, Dordrecht: Kluwer Acad. Publ., pp. 307–327, arXiv:math/0303204, MR 2000728
• Spiridonov, V. P. (2003), "Theta hypergeometric integrals", Rossiĭskaya Akademiya Nauk. Algebra i Analiz, 15 (6): 161–215, arXiv:math/0303205, doi:10.1090/S1061-0022-04-00839-8, MR 2044635
• 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, doi:10.1070/RM2008v063n03ABEH004533, MR 2479997
• Warnaar, S. Ole (2002), "Summation and transformation formulas for elliptic hypergeometric series", Constructive Approximation. an International Journal for Approximations and Expansions, 18 (4): 479–502, arXiv:math/0001006, doi:10.1007/s00365-002-0501-6, MR 1920282