Ó ³Ò ³ ɵ ²µ ʲÓÉ Éµ Ì µé [4, 12Ä15] µ µ ÒÌ Í ²Ó ÒÌ ËÊ ±Í ÖÌ µ³ É Î ±µ µ É, µ É µ ÒÌ Ô²² É Î ± Ì ËÊ ±Í ³µ Ê²Ö ÒÌ Ëµ ³.
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1 ˆ ˆŠ Œ ˆ ˆ Œ ƒ Ÿ 200, Œ 32,. 7 Š ˆ œ Š ˆˆ ƒˆ ƒ Œ ˆ Š ƒ ˆ ˆ ˆ ˆ Šˆ -ˆ ƒ.. µ µ Ñ Ò É ÉÊÉ Ö ÒÌ ² µ, Ê µ ɵ Ö É µ É Î ±µ Ë ± ³... µ µ²õ µ ³µÉ Ò Ô²² É Î ± É - É ²Ò, µ µ Ð ÕÐ µ µ³ Ò É ²Ò ± Ä ²Ó µ ²² Ì Ä Ì³ ³ µ µ³ Ò É ²Ò ²Ó ƒê É Ë µ. Š ɱµ µ - Ê ÕÉ Ö Í ²Ó Ò ËÊ ±Í, µ É µ Ò µ ÒÎ ÒÌ, ÒÌ ³µ Ê²Ö ÒÌ µ- ³ É Î ± Ì Ö µ Ö Ò Ê± Ò³ É ² ³. Elliptic beta integrals, generalizing one variable AskeyÄWilson and NassrallahÄRahman integrals and multivariable Selberg and Gustafson integrals, are considered. Special functions, built from ordinary, basic and modular hypergeometric series and connected with these integrals, are briey reviewed. Ó ³Ò ³ ɵ ²µ ʲÓÉ Éµ Ì µé [4, 2Ä5] µ µ ÒÌ Í ²Ó ÒÌ ËÊ ±Í ÖÌ µ³ É Î ±µ µ É, µ É µ ÒÌ Ô²² É Î ± Ì ËÊ ±Í ³µ Ê²Ö ÒÌ Ëµ ³. É -ËÊ ±Í Ö 0 xα ( x) β dx =Γ(α)Γ(β)/Γ(α+β), Reα, Reβ>0, ² Î Ò q- ²µ ÕÉ ÊÕ µ²ó É µ Í ²Ó ÒÌ ËÊ ±Í []. [3] µ É µ Ò ³Ò µ Ð ±² Î ± ( ³µ Ê ²Ó Ò ) µ ɵ µ- ²Ó Ò µ² µ³ò. É µ² µ³ò Ò ÕÉ Ö Î q- µ³ É Î ± Ö Ò 4 Φ 3. µµé É É ÊÕÐ Ê ²µ µ ³ µ ± Ò µ µ µ ËÊ ±- Í µ ²Ö É Ö q- É - É ²µ³ ± Ä ²Ó µ. [9, 6] µ É µ µ - ³ É µ ³µ Ê ²Ó ÒÌ µ ɵ µ ²Ó ÒÌ Í µ ²Ó ÒÌ ËÊ ±Í ( ), Ò- ÕÐ Ì Ö Î µ Ï µ Ê µ Ï Ò ² µ Ò 0Φ 9 - Ö Ò µ Ð Ì µ² µ³ò ± Ä ²Ó µ ± Î É Î É ÒÌ ²Ê- Î. µ ³ µ ± µµé É É ÊÕÐ Ò µ µ µ ËÊ ±Í É Ö q- É - É ²µ³ ²² Ì Ä Ì³ [8,9], ±µéµ Ò Î É ² Ö ³Ò³ µ Ð ³ É - É ²µ³, µ Ê ± ÕÐ ³ ÒÎ ² ³± Êɵ³. ² Î Ò µ µ Ð Ö ²Ó µ Ä Ì³ Ò² µ É µ Ò [3, 5]. µ- ÒÌ, [5] Ò Ô²² É Î ± µ µ Ð Ö ËÊ ±Í ²Ó µ ± É µ ³ µ µ ɵ µ ²Ó µ É, Ò ÕÐ Ö Î Í ²Ó Ò µ - Ò ÕÐ Ö 2 E ³µ Ê²Ö Ò µ³ É Î ± Ö. Ö Ò É ±µ µ É, µ ² ÕÐ ³µ Ê²Ö µ ³³ É, Ò² Ò Ê Ò³ ± ² ³
2 ˆ œ Š ˆˆ ƒˆ ƒ Œ ˆ Š ƒ ˆ 89 [5]. Ð Ö É µ Ö Ô²² É Î ± Ì µ³ É Î ± Ì Ö µ ˵ ³Ê² µ- [4] ( Î É µ É, Ôɵ µé Ò µ²ó Ê ³Ò Ó µ µ - Î Ö ²Ö E- Ö µ ). ± ²Ó µ ÉÓ Ô²² É Î ± Ì ² Ê É Éµ µ, Îɵ µ² µ Ð, µ Ê ± ÕÐ ±µ Î µ- µ É Ò µ ɵ µ µ µ Ö ± ± Î É µ ÕÐ µ µ ɵ, µ Ö Ò ÒÉÓ Ö Ò³ ± Ô²² É Î ± ³ ɱ ³ [5]. µ- ɵ ÒÌ, Ô²² É Î ±µ µ µ Ð ³ É Ò ÒÌ Ì³ [9] µ É µ µ [3]. ²µ µ ³ µ ± µ- µé É É ÊÕÐ µ²õé µ Ò µ ³ Ò É Ö Ô²² É Î ± ³ É - É ²µ³, ÒÎ ² Ò³ [2] ( ³. ). -É ÉÓ Ì, Ï µ± ±², Ö ²ÖÕÐ Ì Ö ³µ Ê ²Ó Ò³ Ò- ÕÐ Ì Ö Î µ ÒÎ Ò 9 F 8, Ò 0 Φ 9 Ô²² É Î ± 2 E - µ³ É Î ± Ö Ò, µ É µ [5]. Ôɵ³ ±² Ê Ê µ 9F 8 - Ö µ µ ± É µ ̵ ³µ ÉÓ Ï Ö ² Î ±µ µ Ê Ö Ö- ɵ É ²Ö É ² Ö µ³ É Î ± Ì Ö µ. É ËÊ ±Í Ò² µ²êî Ò µ³µðóõ ±µéµ µ µ Ô É Î ±µ µ µ ̵ ± Í ²Ó Ò³ ËÊ ±Í Ö³, µ µ µ µ µ µ []. Ôɵ³ µ ̵ - Í ²Ó Ò ËÊ ±Í µ ²ÖÕÉ Ö ± ± ËÊ ±Í, µí µ Ò ³µ µ- µ Ò³ Ê±Í Ö³ Í µî ± ±É ²Ó ÒÌ µ µ ²Ö ² ÒÌ ±É ²Ó ÒÌ Î. ²ÊÎ ³ É É Ö ±µéµ µ É Ìβ µ ±Ê É µ µµé µï ²Ö µ² µ³µ Ó Ê ³µ ³ µ µ ɵ µ ²Ó- µ É, µ Ð ± É Î ÊÕ ³µ ÉÓ µé ʳ É µ² µ³µ. É ³ ̵ É Ö Í µî± µ µ, ²µ Î ÒÌ µ µ Ö³ Š ɵË- Ë ²Ö ƒ µ ³Ê ²Ö µ ÒÎ ÒÌ µ ɵ µ ²Ó ÒÌ µ² µ³µ, ³ ÖÕÐ Ì Ëµ ³Ò Î ²Ó µ µ ±Ê É µ µ µµé µï Ö. ɵ µ É ± ² - µ Í µî± Ê, µ µ Ð ÕÐ Í µî±ê µ Ò ± É Ò³ ³ ³. ² µ²õé ÒÌ ³³ É Ôɵ Í µî± µ µ²ö É É ³µ µ µ - ÊÕ Ê±Í Õ, µ ÖÐÊÕ ± ±µéµ µ³ê ² µ³ê ËÊ ±Í µ ²Ó µ³ê Ê Õ. ɵ Ê ³ É Ô²² É Î ± Ï Ö, ±µéµ Ò µ- ÖÉ ± Ö µ³ê Ï Õ Î ²Ó µ µ ±Ê É µ µ µµé µï Ö É ³ Ì 2E - Ö µ. ± Ò É Ö, Îɵ µ³ ³µ ³µ Ê²Ö ÒÌ Ï Ò ÒÌ Ì³ ÊÐ É Ê É Ì µ µ Ð É ³Ê ³µ Ê ²Ó ÒÌ ³ µ³µ Ë ÒÌ ËÊ ±Í, µ ɵ µ ²Ó ÒÌ µ ʳ ± É Ò³ ± ³ µ µ ËÊ ±- Í, µ ²ÖÕÐ Ö É ³ Ô²² É Î ± ³ É - É ²µ³ [3]. µ - ³µ µ, Îɵ ÔÉ ËÊ ±Í É ²ÖÕÉ µ µ µ² µ ÐÊÕ É ³Ê ±² - Î ± Ì Í ²Ó ÒÌ ËÊ ±Í É µ µ ɵ µ ²Ó ÒÌ ËÊ ±Í É ³ É ÉÊ µ³, ±µéµ Ò ³ ÕÉ µ² µ³ò ± Ä ²Ó µ É µ µ ɵ µ- ²Ó ÒÌ µ² µ³µ. É ³ µ² µ µ µ, ± Î É ²²Õ É Í, É Ê±ÉÊ Ê Ô²² É Î ±µ µ É - É ² Å µ² µ Ð µ É µ µ ɵÖÐ ³Ö É ² É ±µ µ É. Ê ÉÓ p, q C p, q <. µ Î ³ (a; p) = n=0 ( apn ), (a; p) s =(a; p) /(ap s ; p), (a,...,a k ; p) =(a ; p)... (a k ; p). -
3 90 ˆ ˆ.. ² ³ ÉÔÉ -ËÊ ±Í Õ Ÿ±o ± ± θ(z; p) =(z,pz ; p). µ ² É µ - É a³ θ(pz; p) =θ(z ; p) = z θ(z; p). ²² É Î ± Ö ³³ -ËÊ ±Í Ö Γ(z; p, q) µ ²Ö É Ö µ³µðóõ µ ÒÌ ±µ Î ÒÌ µ [0]: Γ(z; p, q) = j,k=0 z p j+ q k+ zp j q k, Γ(z;0,q)= (z; q) () Ê µ ² É µ Ö É Ê Ö³ Γ(qz; p, q) =θ(z; p)γ(z; p, q), Γ(pz; p, q) =θ(z; q) Γ(z; p, q), ±µéµ Ò µ ² ÒÌ ² Ì µ ÖÉ Ö ± Ê Õ ²Ö É - É µ ³³ -ËÊ ±Í. µ Î ³ É ± θ(a,...,a n ; p) = n i= θ(a i; p), Γ(a,...,a n ; p, q) = n i= Γ(a i; p, q). µ ³. ²² É Î ±µ Ï ±µ ÉÊ µ µ q- É - É ² ²² Ì Ä Ì³ [8, 9] ² Ô²² É Î ± Ö É -ËÊ ±Í Ö ³ É (z) dz 2πi T z = 2 0 m<s 4 Γ(t mt s ; p, q) (q; q) (p; p) 4 m=0 Γ(At m ; p, q), (2) A = 4 m=0 t m (z) = 4 m=0 Γ(zt m,z t m ; p, q) Γ(z 2,z 2,zA,z A; p, q). (3) Ó T µ µ Î É µ²µ É ²Ó µ µ É µ ÊÕ Î ÊÕ µ± Ê µ ÉÓ, ±µ³ ² ± Ò ³ É Ò t m Ê µ ² É µ ÖÕÉ µ Î Ö³ t m <, pq < A. É µ (2) µ± Ò É Ö ³ ɵ µ³ ËÊ ±Í µ ²Ó ÒÌ Ê, ³ - Ò³ [2] ²Ö ÒÎ ² Ö q- É - É ² [8, 9], µ²êî ÕÐ µ Ö (2) p = 0. ²Ö µ Ð µ µ²µ Ö ³ É µ p, q t m ËÊ ±Í Ö (z) µ É ÊÉ T ɵ²Ó±µ µ ÉÒ µ²õ Ò, ̵ ÖÐ Ö ÉµÎ± Ì t m p j q k, A p j+ q k+ j, k N. ² É Î µ ÉÓ (z) µ µ²ö É ³ ÉÓ T ²Õ µ ±µ ÉÊ C, µì ÉÒ ÕÐ ÉµÉ ³ É Î ± µ µ²õ µ. Ôɵ³ Î É ² ³ Ö É Ö, ² Î ÉÓ Ê± ÒÌ µ²õ µ Ê É ² ÉÓ T. Ê ÉÓ f l,r (t 0,t ) Å ² Ö Ö Î É (2) ± ± ËÊ ±Í Ò ² ÒÌ ³ É µ t 0,t. µ ³ 2. Ê ±Í f l,r (t 0,t ) Ê µ ² É µ ÖÕÉ q- µ É µ³ê Ê - Õ t θ(t A, t A; p)f l,r(qt 0,t ) t 0 θ(t 0 A, t 0 A; p)f l,r(t 0,qt )= = t θ(t 0 t,t 0t ; p)f l,r (t 0,t ) (4) Ê Õ, µ²êî ÕÐ ³Ê Ö µ É µ ±µ p q. Ôɵ³ - ³ É Ò t m µ Î Ò Éµ²Ó±µ Ê ²µ ³ Ê²Ö µ É ËÊ ±Í f l,r, ̵- ÖÐ Ì Ê Ö.
4 ˆ œ Š ˆˆ ƒˆ ƒ Œ ˆ Š ƒ ˆ 9 Ö ²Ö Γ(z; p, q) µ µ²öõé µ ÉÓ, Îɵ ËÊ ±Í Ö (z) (3) Ê µ- ² É µ Ö É (4) ²Ö µ µ²ó ÒÌ z ² µ Ö É µ³ê É Ìβ µ³ê ɵ - É Ê θ(xw,x/w,yz,y/z; p) θ(xz,x/z,yw,y/w; p) =yw θ(xy,x/y,wz,w/z; p). µôéµ³ê f l (t 0,t ) ± ± É ² µé (z) µ z É ± Ê µ ² É µ Ö É (4). ²µ- Î µ µ Ö É Ö, Îɵ Ê ²Ö f r (t 0,t ) É ± µ É Ö ± ʱ µ³ê ɵ É Ê. ± ³ µ µ³, µ É ²µ Ó µ ÉÓ, Îɵ f l,r É ²ÖÕÉ µ µ ɵ Ï (4). ɳ É ³, Îɵ É µ µî Ö ³³ É Ö µ µ²ö É ³ ÉÓ t 0 t Ê (4) µ µ²ó ÊÕ Ê ³ É µ t m. µ²µ ³ t 2 = q /2,t 4 = q /2 t,t 3. Ôɵ³ A qt 0, p < t 0 <, É µ (2) ÊÍ Ê É Ö ± É ²Ê 2πi T θ(pz 2 ; p 2 ) dz θ(zt 0,z t 0 ; p) z = (p; p) 2 θ(t 2 0 ; p2 ). (5) ÉµÉ É ² ÒÎ ²Ö É Ö É ±. µ É ³ z =e iϕ ³ ʲ µ ±µôë- Ë Í É ²µ Ö 2π- µ Î ±µ ËÊ ±Í (e iϕ ) Ö Ê Ó µϕ. µ³µðóõ ʳ³Ò ³ Ê ²Ö Ê Éµ µ µ q- µ³ É Î ±µ µ Ö Ψ µé µï θ-ëê ±Í µ ±µ³ É ² (5) ² É Ö µ µ ²µ µ ± Ö µ É Ö³ z. µ ² ÖÉ Ö É ² µ ϕ ÔÉµÉ Ö É µ É Ö µ µ± É Ò³ É ²Ö É µ µ Î É Ò ²ÊΠʳ³ Ê- ³µ µ 2 Ψ 2 - Ö [2]. ˆ µ²ó ÊÖ (4), ³µ µ µ± ÉÓ ( ²µ Î µ [2]), Îɵ É µ (2) Ò- µ² Ö É Ö ²Ö µ µ²ó µ µ t 0, p < t 0 < t = q k,t 2 = q k2+/2,t 3 = q k3,t 4 = q k4+/2 ±µéµ ÒÌ µ²µ É ²Ó ÒÌ Í ²ÒÌ Î ² Ì k i. ²µ- Ö Ö Ò ²µ µ p ² É Î ± µ µ² Ö µ k i µ± Ò ÕÉ, Îɵ É µ (2) Ò µ² Ö É Ö ²Ö µ µ²ó ÒÌ µ Ê É ³ÒÌ Î - ³ É µ t m [2]. [4] ²µ Ò É ³ µ µ³ ÒÌ µ µ Ð Ô²² É Î ±µ µ É - É ² (2), µ ²ÖÕÐ Ì Ô²² É Î ±o Ï É ² ²Ó []. É É ²Ò µ µ Ð ÕÉ ² Î Ò Éµ É Œ ± µ ²Ó Ä Œµ, ÕÐ ÊÕ µ²ó É µ µ ɵ µ ²Ó ÒÌ µ² µ³µ ³ µ- Ì ³ ÒÌ [7], ³ µ µ³ Ò q- É - É ²Ò ƒê É Ë µ [6]. É - É µ µ²µ ÉÓ, Îɵ µ µ ²ÖÕÉ É ± ³ Ê ²Ö ±µéµ µ µ - ³ É µ ɵ µ ²Ó ÒÌ ËÊ ±Í ³ µ Ì ³ ÒÌ, ±²ÕÎ ÕÐ Ì Ö ± ± ËÊ ±Í [3, 5], É ± µ² µ³ò Œ ± µ ²Ó [7]. ˆ Š ˆ. Andrews G. E., Askey R., Roy R. Special Functions // Encyclopedia of Mathematics and its Applications. Cambridge, 999. V. 7.
5 92 ˆ ˆ.. 2. Askey R. Beta Integrals in Ramanujan's Papers, His Unpublished Work and Further Examples // Ramanujan Revisited / Eds. G. E. Andrews et al. Boston, 988. P. 56Ä Askey R., Wilson J. Some Basic Hypergeometric Orthogonal Polynomials that Generalize Jacobi Polynomials // Mem. Amer. Math. Soc V van Diejen J. F., Spiridonov V. P. An Elliptic MacdonaldÄMorris Conjecture and Multiple Modular Hypergeometric Sums // Math. Res. Lett. V P. 729Ä746; Elliptic Selberg Integrals // Internat. Math. Res. Notices 200. No. 20. P. 083Ä0; Modular Hypergeometric Residue Sums of Elliptic Selberg Integrals // Lett. Math. Phys. (to appear). 5. Frenkel I. B., Turaev V. G. Elliptic Solutions of the YangÄBaxter Equation and Modular Hypergeometric Functions // The ArnoldÄGelfand Mathematical Seminars. Boston, 997. P. 7Ä Gustafson R. A. Some q-beta Integrals on SU(n) and Sp(n) that Generalize the AskeyÄWilson and Nassrallah-Rahman Integrals // SIAM J. Math. Anal V. 25. P. 44Ä Macdonald I. G. Constant Term Identities, Orthogonal Polynomials, and Afˇne Hecke Algebras // Doc. Math DMV Extra Volume ICM I. P. 303Ä Nassrallah B., Rahman M. Projection Formulas, a Reproducing Kernel and a Generating Function for q-wilson Polynomials // SIAM J. Math. Anal V. 6. P. 86Ä Rahman M. An Integral Representation of a 0 φ 9 and Continuous Bi-Orthogonal 0 φ 9 Rational Functions // Can. J. Math V. 38. P. 605Ä Ruijsenaars S. N. M. First Order Analytic Difference Equations and Integrable Quantum Systems // J. Math. Phys V. 38. P. 069Ä46.. Spiridonov V. P. The Factorization Method, Self-Similar Potentials and Quantum Algebras // Proc. of the NATO ASI Special Functions Tempe, USA, May 29 Ä June 9, 2000 / Eds. J. Bustoz, M. E. H. Ismail, S. K. Suslov. Dordrecht, 200. P. 335Ä Spiridonov V. P. An Elliptic Beta Integral // Proc. of the Fifth Intern. Conf. on Difference Equations and Applications, Temuco, Chile, Jan. 3Ä7, 2000 / Eds. S. Elaydi, J. Fenner-Lopez, G. Ladas. Gordon and Breach (to appear); µ µ.. Ô²² É Î ±µ É -ËÊ ±Í // Ì ³ É. ʱ T. 56(). C. 8Ä Spiridonov V. P. Elliptic Beta Integrals and Special Functions of Hypergeometric Type // Proc. of the NATO ARW Integrable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory, Kiev, Ukraine, Sept. 25Ä30, 2000 / Eds. G. von Gehlen, S. Pakuliak. Dordrecht, 200. P. 305Ä Spiridonov V. P. Theta Hypergeometric Series // Proc. of the NATO ASI Asymptotic Combinatorics with Applications to Mathematical Physics, St. Petersburg, July 9Ä23, 200 (to appear). 5. Spiridonov V. P., Zhedanov A. S. Spectral Transformation Chains and Some New Biorthogonal Rational Functions // Commun. Math. Phys V. 20. P. 49Ä83; Classical Biorthogonal Rational Functions on Elliptic Grids // C. R. Math. Rep. Acad. Sci. Canada V. 22(2). P. 70Ä76; Generalized Eigenvalue Problem and a New Family of Rational Functions Biorthogonal on Elliptic Grids // Proc. of the NATO ASI Special Functions-2000, Tempe, USA, May 29ÄJune 9, / Eds. J. Bustoz, M. E. H. Ismail, S. K. Suslov. Dordrecht, 200. P. 365Ä388; µ µ.., µ A. C. ƒ µ³ É Î ± µ ɵ µ ²Ó Ò Í µ ²Ó Ò ËÊ ±Í // Ì ³ É. ʱ T. 54(2). C. 73Ä Wilson J. A. Orthogonal Functions from Gram Determinants // SIAM J. Math. Anal. 99. V. 22. P. 47Ä55.
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