Elliptic hypergeometric series

Short description: 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) in their study of elliptic 6-j symbols.

For surveys of elliptic hypergeometric series see (Gasper Rahman), (Spiridonov 2008) or (Rosengren 2016).

Definitions

The q-Pochhammer symbol is defined by

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

$(a_1,a_2,\dots ,a_m;q{)}_n=(a_1;q{)}_n(a_2;q{)}_n\dots (a_m;q{)}_n.$

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

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

$\theta (x_1,...,x_m;p)=\theta (x_1;p)...\theta (x_m;p)$

The elliptic shifted factorial is defined by

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

$(a_1,...,a_m;q,p{)}_n=(a_1;q,p{)}_n\cdots (a_m;q,p{)}_n$

The theta hypergeometric series _r+1E_r is defined by

${}_{r+1}{E}_r(a_1,...a_{r+1};b_1,...,b_r;q,p;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a_1,...,a_{r+1};q;p{)}_n}{(q,b_1,...,b_r;q,p{)}_n}{z}^n$

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

${}_{r+1}{V}_r(a_1;a_6,a_7,...a_{r+1};q,p;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{\theta (a_1{q}^{2n};p)}{\theta (a_1;p)}\frac{(a_1,a_6,a_7,...,a_{r+1};q;p{)}_n}{(q,a_{1}q/a_6,a_{1}q/a_7,...,a_{1}q/a_{r+1};q,p{)}_n}(qz{)}^n$

The bilateral theta hypergeometric series _rG_r is defined by

${}_r{G}_r(a_1,...a_r;b_1,...,b_r;q,p;z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{(a_1,...,a_r;q;p{)}_n}{(b_1,...,b_r;q,p{)}_n}{z}^n$

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

${\theta }_1(x,q)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^n{q}^{(n+1/2{)}^2}{e}^{(2n+1)ix}$

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

${}_{r+1}{e}_r(a_1,...a_{r+1};b_1,...,b_r;\sigma ,\tau ;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{[a_1,...,a_{r+1};\sigma ;\tau {]}_n}{[1,b_1,...,b_r;\sigma ,\tau {]}_n}{z}^n$

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

${}_{r+1}{v}_r(a_1;a_6,...a_{r+1};\sigma ,\tau ;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{[a_1+2n;\sigma ,\tau ]}{[a_1;\sigma ,\tau ]}\frac{[a_1,a_6,...,a_{r+1};\sigma ,\tau {]}_n}{[1,1+a_1-a_6,...,1+a_1-a_{r+1};\sigma ,\tau {]}_n}{z}^n$

Further reading

• Spiridonov, V. P. (2013). "Aspects of elliptic hypergeometric functions". in Berndt, Bruce C.. 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. 20. Ramanujan Mathematical Society. pp. 347–361. ISBN 9789380416137. Bibcode: 2013arXiv1307.2876S. 
• Rosengren, Hjalmar (2016). "Elliptic Hypergeometric Functions". arXiv:1608.06161 [math.CA].

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, https://books.google.com/books?id=Eic6prpQ_VwC&pg=PA171 
• 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 
• 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, Bibcode: 2003math......3204S 
• Spiridonov, V. P. (2003), "Theta hypergeometric integrals", Rossiĭskaya Akademiya Nauk. Algebra i Analiz 15 (6): 161–215, doi:10.1090/S1061-0022-04-00839-8, Bibcode: 2003math......3205S 
• 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, Bibcode: 2008RuMaS..63..405S 
• Warnaar, S. Ole (2002), "Summation and transformation formulas for elliptic hypergeometric series", Constructive Approximation 18 (4): 479–502, doi:10.1007/s00365-002-0501-6