Elliptic hypergeometric series

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

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, 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

Sequences and series

Integer sequences

Basic	Arithmetic progression Geometric progression Harmonic progression Square number Cubic number Factorial Powers of two Powers of three Powers of 10
Advanced (list)	Complete sequence Fibonacci sequence Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number array

Properties of sequences

Properties of series

Series	Alternating Convergent Divergent Telescoping
Convergence	Absolute Conditional Uniform

Explicit series

Convergent	1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/2^s + 1/3^s + ... (Riemann zeta function)
Divergent	1 + 1 + 1 + 1 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) 1 + 2 + 3 + 4 + ⋯ 1 − 2 + 3 − 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ Infinite arithmetic series 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)