Essays on the theory of elliptic hypergeometric functions

V. P. Spiridonov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 63 (2008) no. 3, pp. 405-472

Résumé

This is a brief survey of the main results of the theory of elliptic hypergeometric functions — a new class of special functions of mathematical physics. A proof is given of the most general known univariate exact integration formula generalizing Euler's beta integral. It is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. The known elliptic beta integrals on root systems are listed, and symmetry transformations are considered for the corresponding higher-order elliptic hypergeometric functions.

Bibliographie

[1] G. E. Andrews, R. Askey, R. Roy, Special functions, Encyclopedia Math. Appl., 71, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[2] I. M. Gel'fand, M. I. Graev, V. S. Retakh, “General hypergeometric systems of equations and series of hypergeometric type”, Russian Math. Surveys, 47:4 (1992), 1–88 | DOI | MR | Zbl

[3] G. Gasper, M. Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia Math. Appl., 96, Cambridge Univ. Press, Cambridge, 2004 | MR | Zbl

[4] E. K. Sklyanin, L. A. Takhtadzhyan, L. D. Faddeev, “Quantum inverse problem method. I”, Theoret. Math. Phys., 40:2 (1979), 688–706 | DOI | MR

[5] L. A. Takhtadzhan, L. D. Faddeev, “The quantum method of the inverse problem and the Heisenberg $XYZ$ model”, Russian Math. Surveys, 34:5 (1979), 11–68 | DOI | MR

[6] R. J. Baxter, “Partition function of the eight-vertex lattice model”, Ann. Physics, 70:1 (1972), 193–228 | DOI | MR | Zbl

[7] E. Date, M. Jimbo, A. Kuniba, T. Miwa, M. Okado, “Exactly solvable SOS models, II. Proof of the star-triangle relation and combinatorial identities”, Conformal field theory and solvable lattice models (Kyoto, 1986), Adv. Stud. Pure Math., 16, Academic Press, Boston, MA, 1988, 17–122 | MR | Zbl

[8] I. B. Frenkel, V. G. Turaev, “Elliptic solutions of the Yang–Baxter equation and modular hypergeometric functions”, The Arnold–Gelfand mathematical seminars, Birkhäuser, Boston, MA, 1997, 171–204 | MR | Zbl

[9] V. P. Spiridonov, A. S. Zhedanov, “Spectral transformation chains and some new biorthogonal rational functions”, Comm. Math. Phys., 210:1 (2000), 49–83 | DOI | MR | Zbl

[10] V. P. Spiridonov, “Theta hypergeometric series”, Asymptotic combinatorics with application to mathematical physics (St. Petersburg, 2001), NATO Sci. Ser. II Math. Phys. Chem., 77, Kluwer, Dordrecht, 2002, 307–327 ; arXiv: math/0303204 | MR | Zbl

[11] V. P. Spiridonov, “Theta hypergeometric integrals”, Algebra i analiz, 15:6 (2003), 161–215 ; V. P. Spiridonov, “Theta hypergeometric integrals”, St. Petersburg Math. J., 15:6 (2004), 929–967 ; arXiv: math/0303205 | MR | Zbl | DOI

[12] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, vol. I, McGraw-Hill, New York–Toronto–London, 1953 | MR | Zbl | Zbl

[13] E. W. Barnes, “On the theory of the multiple gamma function”, Cambr. Trans., 19 (1904), 374–425 | Zbl

[14] F. H. Jackson, “The basic gamma-function and the elliptic functions”, Proc. Roy. Soc. Lond. Ser. A, 76:508 (1905), 127–144 | DOI | Zbl

[15] T. Shintani, “On a Kronecker limit formula for real quadratic field”, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24:1 (1977), 167–199 | MR | Zbl

[16] N. Kurokawa, “Multiple sine functions and Selberg zeta functions”, Proc. Japan Acad. Ser. A Math. Sci., 67:3 (1991), 61–64 | DOI | MR | Zbl

[17] L. D. Faddeev, “Discrete Heisenberg–Weyl group and modular group”, Lett. Math. Phys., 34:3 (1995), 249–254 | DOI | MR | Zbl

[18] S. N. M. Ruijsenaars, “First order analytic difference equations and integrable quantum systems”, J. Math. Phys., 38:2 (1997), 1069–1146 | DOI | MR | Zbl

[19] M. Jimbo, T. Miwa, “Quantum KZ equation with $|q|=1$ and correlation functions of the $XXZ$ model in the gapless regime”, J. Phys. A, 29:12 (1996), 2923–2958 | DOI | MR | Zbl

[20] V. Tarasov, A. Varchenko, “Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups”, Astérisque, 1997, no. 246, 1–135 ; arXiv: q-alg/9703044 | MR | Zbl

[21] L. D. Faddeev, R. M. Kashaev, A. Yu. Volkov, “Strongly coupled quantum discrete Liouville theory. I: Algebraic approach and duality”, Commun. Math. Phys., 219:1 (2001), 199–219 | DOI | MR | Zbl

[22] B. Ponsot, J. Teschner, “Clebsch–Gordan and Racah–Wigner coefficients for a continuous series of representations of $U_q(sl(2,\mathbb{R}))$”, Comm. Math. Phys., 224:3 (2001), 613–655 | DOI | MR | Zbl

[23] S. Kharchev, D. Lebedev, M. Semenov-Tian-Shansky, “Unitary representations of $U_q\bigl(sl(2,\mathbb R)\bigr)$, the modular double and the multiparticle $q$-deformed Toda chains”, Comm. Math. Phys., 225:3 (2002), 573–609 | DOI | MR | Zbl

[24] A. Yu. Volkov, “Noncommutative hypergeometry”, Comm. Math. Phys., 258:2 (2005), 257–273 | DOI | MR | Zbl

[25] G. Felder, A. Varchenko, “The elliptic gamma function and $SL(3,\mathbb Z)\ltimes\mathbb Z^3$”, Adv. Math., 156:1 (2000), 44–76 | DOI | MR | Zbl

[26] E. Friedman, S. Ruijsenaars, “Shintani–Barnes zeta and gamma functions”, Adv. Math., 187:2 (2004), 362–395 | DOI | MR | Zbl

[27] A. Narukawa, “The modular properties and the integral representations of the multiple elliptic gamma functions”, Adv. Math., 189:2 (2004), 247–267 | DOI | MR | Zbl

[28] E. M. Rains, “Limits of elliptic hypergeometric integrals”, Ramanujan J., to appear; , 2006 arXiv: math/0607093

[29] V. P. Spiridonov, “On the elliptic beta function”, Russian Math. Surveys, 56:1 (2001), 185–186 | DOI | MR | Zbl

[30] V. P. Spiridonov, “A Bailey tree for integrals”, Theoret. and Math. Phys., 139:1 (2004), 536–541 | DOI | MR

[31] V. P. Spiridonov, Ellipticheskie gipergeometricheskie funktsii, Dis. $\dots$ dokt. fiz.-matem. nauk, LTF OIYaI, Dubna, 2004

[32] J. F. van Diejen, V. P. Spiridonov, “An elliptic Macdonald–Morris conjecture and multiple modular hypergeometric sums”, Math. Res. Lett., 7:5–6 (2000), 729–746 | MR | Zbl

[33] J. F. van Diejen, V. P. Spiridonov, “Elliptic Selberg integrals”, Internat. Math. Res. Notices, 2001, no. 20, 1083–1110 | DOI | MR | Zbl

[34] E. M. Rains, “Transformations of elliptic hypergeometric integrals”, Ann. of Math. (to appear)

[35] E. M. Rains, “$BC_n$-symmetric Abelian functions”, Duke Math. J., 135:1 (2006), 99–180 | DOI | MR | Zbl

[36] V. P. Spiridonov, S. O. Warnaar, “Inversions of integral operators and elliptic beta integrals on root systems”, Adv. Math., 207:1 (2006), 91–132 | DOI | MR | Zbl

[37] S. N. M. Ruijsenaars, “On Barnes' multiple zeta and gamma functions”, Adv. Math., 156:1 (2000), 107–132 | DOI | MR | Zbl

[38] J. F. van Diejen, V. P. Spiridonov, “Unit circle elliptic beta integrals”, Ramanujan J., 10:2 (2005), 187–204 | DOI | MR | Zbl

[39] M. Rahman, “An integral representation of a ${}_{10}\phi_9$ and continuous bi-orthogonal ${}_{10}\phi_9$ rational functions”, Canad. J. Math., 38:3 (1986), 605–618 | MR | Zbl

[40] B. Nassrallah, M. Rahman, “Projection formulas, a reproducing kernel and a generating function for $q$-Wilson polynomials”, SIAM J. Math. Anal., 16:1 (1985), 186–197 | DOI | MR | Zbl

[41] R. Askey, J. Wilson, “Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials”, Mem. Amer. Math. Soc., 54:319 (1985) | MR | Zbl

[42] R. Askey, “Beta integrals in Ramanujan's papers, his unpublished work and further examples”, Ramanujan revisited (Urbana-Champaign, IL, 1987), Academic Press, Boston, MA, 1988, 561–590 | MR | Zbl

[43] V. P. Spiridonov, “Short proofs of the elliptic beta integrals”, Ramanujan J., 13:1–3 (2007), 265–283 ; arXiv: math/0408369 | DOI | MR | Zbl

[44] H. S. Wilf, D. Zeilberger, “An algorithmic proof theory for hypergeometric (ordinary and "$q$") multisum/integral identities”, Invent. Math., 108:3 (1992), 575–633 | DOI | MR | Zbl

[45] H. Rosengren, “A proof of a multivariable elliptic summation formula conjectured by Warnaar”, $q$-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000), Contemp. Math., 291, Amer. Math. Soc., Providence, RI, 2001, 193–202 ; arXiv: math/0101073 | MR | Zbl

[46] V. P. Spiridonov, A. S. Zhedanov, “To the theory of biorthogonal rational functions”, Sūrikaisekikenkyūsho Kōkyūroku, 2003, no. 1302, 172–192 | MR

[47] M. Schlosser, “Elliptic enumeration of nonintersecting lattice paths”, J. Combin. Theory Ser. A, 114:3 (2007), 505–521 | DOI | MR | Zbl

[48] M. Schlosser, A Taylor expansion theorem for an elliptic extension of the Askey–Wilson operator, arXiv: 0803.2329

[49] W. Chu, C. Jia, “Abel's method on summation by parts and theta hypergeometric series”, J. Combin. Theory Ser. A (to appear)

[50] J. V. Stokman, “Hyperbolic beta integrals”, Adv. Math., 190:1 (2005), 119–160 | DOI | MR | Zbl

[51] L. J. Slater, Generalized hypergeometric functions, Cambridge Univ. Press, Cambridge, 1966 | MR | Zbl

[52] V. P. Spiridonov, “An elliptic incarnation of the Bailey chain”, Int. Math. Res. Not., 2002, no. 37, 1945–1977 | DOI | MR | Zbl

[53] S. O. Warnaar, “Summation and transformation formulas for elliptic hypergeometric series”, Constr. Approx., 18:4 (2002), 479–502 | DOI | MR | Zbl

[54] S. O. Warnaar, “Summation formulae for elliptic hypergeometric series”, Proc. Amer. Math. Soc., 133:2 (2005), 519–527 | DOI | MR | Zbl

[55] G. Gasper, M. Schlosser, “Summation, transformation, and expansion formulas for multibasic theta hypergeometric series”, Adv. Stud. Contemp. Math. (Kyungshang), 11:1 (2005), 67–84 ; arXiv: math/0505215 | MR | Zbl

[56] H. Rosengren, M. Schlosser, “On Warnaar's elliptic matrix inversion and Karlsson–Minton-type elliptic hypergeometric series”, J. Comput. Appl. Math., 178:1–2 (2005), 377–391 | DOI | MR | Zbl

[57] A. Zhedanov, Elliptic polynomials orthogonal on the unit circle with a dense point spectrum, arXiv: 0711.4696

[58] V. P. Spiridonov, “Elliptic hypergeometric functions and Calogero–Sutherland-type models”, Theoret. Math. Phys., 150:2 (2007), 266–277 | DOI | MR | Zbl

[59] E. M. Rains, V. P. Spiridonov, Determinants of elliptic hypergeometric integrals, arXiv: 0712.4253

[60] S. N. M. Ruijsenaars, “Generalized hypergeometric function satisfying four analytic difference equations of Askey–Wilson type”, Comm. Math. Phys., 206:3 (1999), 639–690 | DOI | MR | Zbl

[61] F. J. van de Bult, E. M. Rains, J. V. Stokman, “Properties of generalized univariate hypergeometric functions”, Comm. Math. Phys., 275:1 (2007), 37–95 ; arXiv: math/0607250 | DOI | MR | Zbl

[62] J. F. van Diejen, “Integrability of difference Calogero–Moser systems”, J. Math. Phys., 35:6 (1994), 2983–3004 | DOI | MR | Zbl

[63] Y. Komori, K. Hikami, “Quantum integrability of the generalized elliptic Ruijsenaars models”, J. Phys. A, 30:12 (1997), 4341–4364 | DOI | MR | Zbl

[64] S. N. M. Ruijsenaars, “Complete integrability of relativistic Calogero–Moser systems and elliptic function identities”, Comm. Math. Phys., 110:2 (1987), 191–213 | DOI | MR | Zbl

[65] V. I. Inozemtsev, “Lax representation with spectral parameter on a torus for integrable particle systems”, Lett. Math. Phys., 17:1 (1989), 11–17 | DOI | MR | Zbl

[66] G. E. Andrews, “Bailey's transform, lemma, chains and tree”, Special functions 2000: current perspective and future directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., 30, Kluwer, Dordrecht, 2001, 1–22 | MR | Zbl

[67] S. O. Warnaar, “Extensions of the well-poised and elliptic well-poised Bailey lemma”, Indag. Math. (N.S.), 14:3–4 (2003), 571–588 | DOI | MR | Zbl

[68] D. M. Bressoud, “A matrix inverse”, Proc. Amer. Math. Soc., 88:3 (1983), 446–448 | DOI | MR | Zbl

[69] X. R. Ma, “An extension of Warnaar's matrix inversion”, Proc. Amer. Math. Soc., 133:11 (2005), 3179–3189 | DOI | MR | Zbl

[70] A. Zhedanov, “Biorthogonal rational functions and the generalized eigenvalue problem”, J. Approx. Theory, 101:2 (1999), 303–329 | DOI | MR | Zbl

[71] M. E. H. Ismail, D. R. Masson, “Generalized orthogonality and continued fractions”, J. Approx. Theory, 83:1 (1995), 1–40 | DOI | MR | Zbl

[72] A. A. Gonchar, “On the speed of rational approximation of some analytic functions”, Math. USSR-Sb., 34:2 (1978), 131–145 | DOI | MR | Zbl | Zbl

[73] A. A. Gonchar, G. López Lagomasino, “On Markov's theorem for multipoint Padé approximants”, Math. USSR-Sb., 34:4 (1978), 449–459 | DOI | MR | Zbl

[74] V. P. Spiridonov, A. S. Zhedanov, “Classical biorthogonal rational functions on elliptic grids”, C. R. Math. Acad. Sci. Soc. R. Can., 22:2 (2000), 70–76 | MR | Zbl

[75] J. A. Wilson, “Orthogonal functions from Gram determinants”, SIAM J. Math. Anal., 22:4 (1991), 1147–1155 | DOI | MR | Zbl

[76] H. Rosengren, “An elementary approach to $6j$-symbols (classical, quantum, rational, trigonometric, and elliptic)”, Ramanujan J., 13:1–3 (2007), 131–166 | DOI | MR | Zbl

[77] D. P. Gupta, D. R. Masson, “Contiguous relations, continued fractions and orthogonality”, Trans. Amer. Math. Soc., 350:2 (1998), 769–808 | DOI | MR | Zbl

[78] D. P. Gupta, D. R. Masson, “Watson's basic analogue of Ramanujan's entry 40 and its generalization”, SIAM J. Math. Anal., 25:2 (1994), 429–440 | DOI | MR | Zbl

[79] V. P. Spiridonov, “Nepreryvnaya biortogonalnost ellipticheskoi gipergeometricheskoi funktsii”, Algebra i analiz, 20:5 (2008), 155–185 ; arXiv: 0801.4137 | MR

[80] H. Rosengren, “Sklyanin invariant integration”, Int. Math. Res. Not., 2004, no. 60, 3207–3232 ; arXiv: math/0405072 | DOI | MR | Zbl

[81] E. K. Sklyanin, “Some algebraic structures connected with the Yang–Baxter equation”, Funct. Anal. Appl., 16:4 (1982), 263–270 | DOI | MR | Zbl

[82] E. K. Sklyanin, “Some algebraic structures connected with the Yang–Baxter equation. Representations of quantum algebras”, Funct. Anal. Appl., 17:4 (1983), 273–284 | DOI | MR | Zbl

[83] Yu. I. Manin, “Sixth Painlevé equation, universal elliptic curve, and mirror of $P^2$”, Geometry of differential equations, Amer. Math. Soc. Transl. Ser. 2, 186, Amer. Math. Soc., Providence, RI, 1998, 131–151 | MR | Zbl

[84] L. D. Faddeev, “Modular double of a quantum group”, Conférence Moshé Flato, vol. I (Dijon, 1999), Math. Phys. Stud., 21, Kluwer, Dordrecht, 2000, 149–156 | MR | Zbl

[85] E. T. Whittaker, G. N. Watson, A course of modern analysis, Cambridge Univ. Press., Cambridge, 1915 | MR | Zbl

[86] H. Rosengren, “Elliptic hypergeometric series on root systems”, Adv. Math., 181:2 (2004), 417–447 | DOI | MR | Zbl

[87] R. A. Gustafson, “Multilateral summation theorems for ordinary and basic hypergeometric series in $U(n)$”, SIAM J. Math. Anal., 18:6 (1987), 1576–1596 | DOI | MR | Zbl

[88] C. Krattenthaler, “The major counting of nonintersecting lattice paths and generating functions for tableaux”, Mem. Amer. Math. Soc., 115:552 (1995) | MR | Zbl

[89] Y. Kajihara, M. Noumi, “Multiple elliptic hypergeometric series. An approach from the Cauchy determinant”, Indag. Math. (N.S.), 14:3–4 (2003), 395–421 | DOI | MR | Zbl

[90] H. Rosengren, M. Schlosser, “Summations and transformations for multiple basic and elliptic hypergeometric series by determinant evaluations”, Indag. Math. (N.S.), 14:3–4 (2003), 483–513 | DOI | MR | Zbl

[91] E. M. Rains, “Recurrences for elliptic hypergeometric integrals”, Rokko Lect. Math., 18 (2005), 183–199; arXiv: math/0504285

[92] H. Rosengren, M. Schlosser, “Elliptic determinant evaluations and the Macdonald identities for affine root systems”, Compos. Math., 142:4 (2006), 937–961 | DOI | MR | Zbl

[93] H. Rosengren, “Sums of triangular numbers from the Frobenius determinant”, Adv. Math., 208:2 (2007), 935–961 | DOI | MR | Zbl

[94] H. Rosengren, An Izergin–Korepin-type identity for the 8VSOS model, with applications to alternating sign matrices, arXiv: 0801.1229

[95] R. A. Gustafson, “Some $q$-beta and Mellin–Barnes integrals with many parameters associated to the classical groups”, SIAM J. Math. Anal., 23:2 (1992), 525–551 | DOI | MR | Zbl

[96] R. A. Gustafson, “Some $q$-beta integrals on $SU(n)$ and $Sp(n)$ that generalize the Askey–Wilson and Nassrallah–Rahman integrals”, SIAM J. Math. Anal., 25:2 (1994), 441–449 | DOI | MR | Zbl

[97] P. J. Forrester, S. O. Warnaar, “The importance of the Selberg integral”, Bull. Amer. Math. Soc. (N.S.), to appear

[98] I. G. Macdonald, “Constant term identities, orthogonal polynomials, and affine Hecke algebras”, Proceedings of the International Congress of Mathematicians Vol. I (Berlin, 1998), Doc. Math., 1998 Extra Vol. I, 303–317 (electronic) | MR | Zbl

[99] G. W. Anderson, “A short proof of Selberg's generalized beta formula”, Forum Math., 3:4 (1991), 415–417 | MR | Zbl

[100] R. A. Gustafson, M. A. Rakha, “$q$-Beta integrals and multivariate basic hypergeometric series associated to root systems of type $A_m$”, Ann. Comb., 4:3–4 (2000), 347–373 | DOI | MR | Zbl

[101] J. F. van Diejen, V. P. Spiridonov, “Modular hypergeometric residue sums of elliptic Selberg integrals”, Lett. Math. Phys., 58:3 (2001), 223–238 | DOI | MR | Zbl

[102] H. Coskun, R. A. Gustafson, “Well-poised Macdonald functions $W_{\lambda}$ and Jackson coefficients $\omega_{\lambda}$ on $BC_n$”, Jack, Hall–Littlewood and Macdonald polynomials, Contemp. Math., 417, Amer. Math. Soc., Providence, RI, 2006, 127–155 | MR | Zbl

[103] R. Y. Denis, R. A. Gustafson, “An $SU(n)$ $q$-beta integral transformation and multiple hypergeometric series identities”, SIAM J. Math. Anal., 23:2 (1992), 552–561 | DOI | MR | Zbl

[104] S. C. Milne, “Multiple $q$-series and $U(n)$ generalizations of Ramanujan's $_1\Psi_1$ sum”, Ramanujan revisited (Urbana-Champaign, IL, 1987), Academic Press, Boston, MA, 1988, 473–524 | MR | Zbl

[105] A. L. Dixon, “Generalisations of Legendre's formula $KE'-(K-E)K'=\frac12\pi$”, Proc. London Math. Soc., 2:1 (1905), 206–224 | DOI | Zbl

[106] V. P. Spiridonov, A. S. Zhedanov, “Elliptic grids, rational functions, and the Padé interpolation”, Ramanujan J., 13:1–3 (2007), 285–310 | DOI | MR | Zbl

[107] A. Zhedanov, “Padé interpolation table and biorthogonal rational functions”, Rokko Lect. Math., 18 (2005), 323–363

[108] T. H. Koornwinder, “Askey–Wilson polynomials for root systems of type $BC$”, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991), Contemp. Math., 138, Amer. Math. Soc., Providence, RI, 1992, 189–204 | MR | Zbl

[109] A. Okounkov, “$BC$-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials”, Transform. Groups, 3:2 (1998), 181–207 | DOI | MR | Zbl

[110] G. Felder, A. Varchenko, “Hypergeometric theta functions and elliptic Macdonald polynomials”, Int. Math. Res. Not., 2004, no. 21, 1037–1055 | DOI | MR | Zbl

[111] H. Sakai, “Rational surfaces associated with affine root systems and geometry of the Painlevé equations”, Comm. Math. Phys., 220:1 (2001), 165–229 | DOI | MR | Zbl

[112] K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, “${}_{10}E_9$ solution to the elliptic Painlevé equation”, J. Phys. A, 36:17 (2003), L263–L272 | DOI | MR | Zbl

[113] M. Ito, “Askey–Wilson type integrals associated with root systems”, Ramanujan J., 12:1 (2006), 131–151 | DOI | MR | Zbl

[114] M. E. H. Ismail, M. Rahman, “The associated Askey–Wilson polynomials”, Trans. Amer. Math. Soc., 328:1 (1991), 201–237 | DOI | MR | Zbl

[115] V. P. Spiridonov, “A multiparameter summation formula for Riemann theta functions”, Jack, Hall–Littlewood and Macdonald polynomials, Contemp. Math., 417, Amer. Math. Soc., Providence, RI, 2006, 345–353 | MR | Zbl

[116] H. Konno, “The vertex-face correspondence and the elliptic $6j$-symbols”, Lett. Math. Phys., 72:3 (2005), 243–258 ; arXiv: math/0503725 | DOI | MR | Zbl

[117] N. I. Akhiezer, Elements of the theory of elliptic functions, Transl. Math. Monogr., 79, Amer. Math. Soc., Providence, RI, 1990 | MR | MR | Zbl

[118] M. Eichler, D. Zagier, The theory of Jacobi forms, Progr. Math., 55, Birkhäuser, Boston, MA, 1985 | MR | Zbl