Elliptic hypergeometric functions and elliptic difference Painlev e - PowerPoint PPT Presentation

Elliptic hypergeometric functions and elliptic difference Painlev´ e equation Masatoshi NOUMI (Kobe University, Japan) ICMP 2018, Montreal, Canada (July 27, 2018) Abstract Elliptic hypergeometric functions are a new class of special functions that have been developed during these two decades. In this talk I will give an overview of various as- pects of elliptic hypergeometric functions with emphasis on connections with integrable systems including the elliptic difference Painlev´ e equation.

[1] Plan of this talk Part 1: Elliptic Hypergeometric Functions Part 2: Elliptic Difference Painlev´ e Equation

[2] Part 1: Elliptic Hypergeometric Functions References for Part 1 [1] M. Ito and M. Noumi: Derivation of a BC n elliptic summation formula via the fundamental invariants, Constr. Approx. 45 (2017), 33–46 (arXiv:1504.07018, 11 pages). [2] M. Ito and M. Noumi: Evaluation of the BC n elliptic Selberg integral via the fundamental invari- ants, Proc. Amer. Math. Soc. 145 (2017), 689–703 (arXiv:1504.07317, 15 pages). [3] M. Ito and M. Noumi: A determinant formula associated with the elliptic hypergeometric integrals of type BC n (in preparation).

[3] q -Hypergeometric integrals of Selberg type 1 ○ Selberg integral (1942) Generalization of the beta integral to a multiple integral involving a power of the difference product (Atle Selberg, 1917–2007): ∫ 1 ∫ 1 n 1 | z i − z j | 2 γ dz 1 · · · dz n ∏ ∏ z α − 1 (1 − z i ) β − 1 · · · i n ! 0 0 i =1 1 ≤ i<j ≤ n n Γ( α + ( j − 1) γ ) Γ( β + ( j − 1) γ ) Γ( jγ ) ∏ = Γ( α + β + ( n + j − 2) γ ) Γ( γ ) j =1 Variations and extensions of this formula, including the cases of integrals of trigono- metric and elliptic fuctions, provide with foundations for a variety of theories of hyper- geometric functions in many variables. • Hypergeometric integral of Selberg type = Selberg integral in the broad sense Integral of powers of polynomials which involves a power of a difference product or a Weyl denominator • Selberg integral in the narrow sense Hypergeometric integral of Selberg type which admits an evaluation formula in terms of the gamma function

[4] ○ q -Hypergeometric integrals of Selberg type z = ( z 1 , . . . , z n ): coordinates of the n -dimensional algebraic torus T n = ( C ∗ ) n There are two types of q -hypergeometric integrals (with base q ∈ C ∗ , | q | < 1): Jackson integrals /infinite multiple series (Aomoto–Ito), versus ordinary integrals over n -cycles in T n (Macdonald) • Jackson integral: With a base point ζ = ( ζ 1 , . . . , ζ n ) ∈ ( C ∗ ) n , the Jackson integral of a function ϕ ( z ) is defined as the infinite multiple series ∫ ζ 1 ∞ ∫ ζ n ∞ ∞ ∞ 1 ϕ ( z 1 , . . . , z n ) d q z 1 · · · d q z n ∑ ∑ ϕ ( q ν 1 ζ 1 , . . . , q ν n ζ n ) . · · · = · · · (1 − q ) n z 1 · · · z n 0 0 ν 1 = −∞ ν n = −∞ In the notation of multi-indices ν = ( ν 1 , . . . , ν n ) ∈ Z n , q ν ζ = ( q ν 1 ζ 1 , . . . , q ν n ζ n ) ∈ ( C ∗ ) n , ∫ ζ ∞ 1 d q z 1 · · · d q z n ∑ ϕ ( ζq ν ) , ϕ ( z ) ω q ( z ) = ω q ( z ) = . (1 − q ) n z 1 · · · z n 0 ν ∈ Z n Sum of the values of ϕ ( z ) over the multiplicative lattice Λ ζ = q Z n ζ ⊂ ( C ∗ ) n . • Ordinary integral over an n -cycle: 1 ϕ ( z 1 , . . . , z n ) dz 1 · · · dz n 1 dz 1 · · · dz n ∫ ∫ ϕ ( z ) ω ( z ) = (2 π √− 1) n , ω ( z ) = (2 π √− 1) n z 1 · · · z n z 1 · · · z n C C Typically, the real torus T n R = {| z 1 | = · · · = | z n | = 1 } is chosen for the n -cycle C .

[5] ○ q -Shifted factorials • q -Shifted factorials: ∞ ( z ; q ) ∞ ∏ (1 − q i z ) , ( z ; q ) ∞ = ( z ; q ) k = ( k ∈ Z ) ( q k z ; q ) ∞ i =0 For k = 0 , 1 , 2 , . . . , 1 ( z ; q ) k = (1 − z )(1 − qz ) · · · (1 − q k − 1 z ) , ( z ; q ) − k = (1 − q − k z )(1 − q − k +1 z ) · · · (1 − q − 1 z ) . q -Shifted factorials are regarded as counterparts of power functions or gamma functions : ( q β z ; q ) ∞ ( q ; q ) ∞ (1 − z ) α − β ; (1 − q ) 1 − s → → Γ( s ) ( q α z ; q ) ∞ ( q s ; q ) ∞ For k ∈ Z or k = ∞ , a product of q -shifted factorials are often abbreviated as ( a 1 , . . . , a r ; q ) k = ( a 1 ; q ) k · · · ( a r ; q ) k .

[6] ○ q -Beta and q -hypergeometric integrals (contour integrals) double sign: f ( z ± 1 ) = f ( z ) f ( z − 1 ) • Askey–Wilson q -beta integral: ( z ± 2 ; q ) ∞ 1 ∫ dz 2 ( abcd ; q ) ∞ 2 π √− 1 z = ( az ± 1 , bz ± 1 , cz ± 1 , dz ± 1 ; q ) ∞ ( q ; q ) ∞ ( ab, ac, ad, bc, bd, cd ; q ) ∞ C C : a closed curve separating the poles accumulating at z = 0 and those at z = ∞ . • Nassrallah–Rahman q -beta integral: Under the condition a 0 a 1 · · · a 5 = q , ∏ 5 ( z ± 2 ; q ) ∞ ( qa − 1 0 z ± 1 ; q ) ∞ 1 dz 2 i =1 ( q/a i a 0 ; q ) ∞ ∫ 2 π √− 1 z = ∏ 5 ∏ ( q ; q ) ∞ 1 ≤ i<j ≤ 5 ( a i a j ; q ) ∞ k =1 ( a k z ± 1 ; q ) ∞ C • Rahman’s q -hypergeometric integral: (Rahman 1986) Under the balancing condition a 0 a 1 · · · a 7 = q 2 , ( z ± 2 ; q ) ∞ i =0 , 7 ( qa − 1 i z ± 1 ; q ) ∞ ∏ ( a i a j ; q ) ∞ · ( q ; q ) ∞ ∫ dz ∏ 4 π √− 1 ∏ 6 z i =1 ( a i z ± 1 ; q ) ∞ C 1 ≤ i<j ≤ 6 ∏ 6 i =1 ( qa i /a 0 ; q ) ∞ ( q/a i a 7 ; q ) ∞ q/a 2 ( ) = 10 W 9 0 ; q/a 0 a 1 , q/a 0 a 2 , . . . , q/a 0 a 7 ; q, q ( q 2 a 2 0 ; q ) ∞ ( a 0 /a 7 ; q ) ∞ ∏ 6 i =1 ( qa i /a 7 ; q ) ∞ ( q/a i a 0 ; q ) ∞ q/a 2 ( ) + 10 W 9 7 ; q/a 1 a 7 , q/a 2 a 7 , . . . , q/a 6 a 7 ; q, q . ( q 2 a 2 7 ; q ) ∞ ( a 7 /a 0 ; q ) ∞ ∞ r 1 − q 2 k a 0 ( a 0 ; q ) k ( a i ; q ) k ∑ ∏ z k ( ) r +3 W r +2 a 0 ; a 1 , . . . , a r ; q, z = 1 − a 0 ( q ; q ) k ( qa 0 /a i ; q ) k k =0 i =1

[7] ○ q -Hypergeometric integral of Selberg type z = ( z 1 , . . . , z n ): coordinates of the algebraic torus T n = ( C ∗ ) n • Gustafson’s q -Selberg integral (1990) [Askey–Wilson] For generic complex parameters a = ( a 1 , . . . , a 4 ) and t , n ( z ± 1 i z ± 1 ( z ± 2 j ; q ) ∞ 1 ∫ i ; q ) ∞ dz 1 · · · dz n ∏ ∏ (2 π √− 1) n ∏ 4 ( tz ± 1 i z ± 1 k =1 ( a k z ± 1 z 1 · · · z n j ; q ) ∞ i ; q ) ∞ C n i =1 1 ≤ i<j ≤ n ( ( t ; q ) ∞ n 2 n n ! ( a 1 a 2 a 3 a 4 t n + i − 2 ; q ) ∞ ) ∏ = ( q ; q ) n ( t i ; q ) ∞ ∏ 1 ≤ k<l ≤ 4 ( t i − 1 a k a l ; q ) ∞ ∞ i =1 The integrand is the weight function for the Koornwinder polynomials ( BC n ). [Nassrallah-Rahman] Under the balancing condition a 0 a 1 a 2 a 3 a 4 a 5 t 2 n − 2 = q 2 , n ( z ± 1 i z ± 1 ( z ± 2 i ; q ) ∞ ( qa − 1 0 z ± 1 j ; q ) ∞ 1 ∫ i ; q ) ∞ dz 1 · · · dz n ∏ ∏ (2 π √− 1) n ∏ 5 ( tz ± 1 i z ± 1 k =1 ( a k z ± 1 z 1 · · · z n j ; q ) ∞ i ; q ) ∞ C n i =1 1 ≤ i<j ≤ n ( ( t ; q ) ∞ n ∏ 5 2 n n ! k =1 ( t 1 − i q/a 0 a k ; q ) ∞ ) ∏ = ( q ; q ) n ( t i ; q ) ∞ ∏ 1 ≤ k<l ≤ 5 ( t i − 1 a k a l ; q ) ∞ ∞ i =1

[8] Elliptic hypergeometric integrals of Selberg type 2 ○ Ruijsenaars’ elliptic gamma function With two (generic) bases p, q ∈ C ∗ , | p | < 1 , | q | < 1, ∞ Γ( z ; p, q ) = ( pq/z ; p, q ) ∞ ∏ (1 − p i q j z ) . , ( z ; p, q ) ∞ = ( z ; p, q ) ∞ i,j =0 It is a meromorphic function on C ∗ with simple poles at z = p − i q − j ( i, j = 0 , 1 , . . . ). ・ Jacobi theta function (in the multiplicative variable): θ ( pz ; p ) = − z − 1 θ ( z ; p ) , θ ( z ; p ) = ( z ; p ) ∞ ( p/z ; p ) ∞ ; θ ( p/z ; p ) = θ ( z ; p ) ・ The elliptic gamma function satisfies the following functional equations: Γ( pq/z ; p, q ) = Γ( z ; p, q ) − 1 Γ( qz ; p, q ) = θ ( z ; p )Γ( z ; p, q ) , ・ In the double sign notation f ( z ± 1 ) = f ( z ) f ( z − 1 ), ( z ± 1 ; p, q ) ∞ 1 = (1 − z ± 1 )( pz ± 1 ; p ) ∞ ( qz ± 1 ; q ) ∞ Γ( z ± 1 ; p, q ) = ( pqz ± 1 ; p, q ) ∞ = − z − 1 ( z, p/z ; p ) ∞ ( z, q/z ; q ) ∞ = − z − 1 θ ( z ; p ) θ ( z ; q ) holomorphic on C ∗ , splits into the product of two theta functions with bases p , q . ・ In the limit as p → 0, 1 θ ( z ; p ) → (1 − z ) , Γ( z ; p, q ) → , Γ( pz ; p, q ) → ( q/z ; q ) ∞ ( z ; q ) ∞

[9] ○ Elliptic hypergeometric integral of Selberg type ( BC n ) • Elliptic beta integral (Spiridonov 2001) Under the balancing condition a 1 · · · a 6 = pq , ∏ 6 k =1 Γ( a k z ± 1 ; p, q ) ( p ; p ) ∞ ( q ; q ) ∞ dz ∫ ∏ 4 π √− 1 z = Γ( a k a l ; p, q ) Γ( z ± 2 ; p, q ) C 1 ≤ k<l ≤ 6 ・ Elliptic extension of the Nassrallah–Rahman q -beta integral ・ Integral version of the Frenkel–Turaev sum • Elliptic hypergeometric integral of Selberg type The following integral is called the BC n elliptic hypergeometric integral of Selberg type : 1 dz 1 · · · dz n ∫ (2 π √− 1) n I n ( a ) = C n Φ( z ; a ) ω ( z ) , ω ( z ) = z 1 · · · z n n ∏ m Γ( tz ± 1 i z ± 1 k =1 Γ( a k z ± 1 j ; p, q ) i ; p, q ) ∏ ∏ Φ( z ; a ) = Γ( z ± 2 Γ( z ± 1 i z ± 1 i ; p, q ) j ; p, q ) i =1 1 ≤ i<j ≤ n a = ( a 1 , . . . , a m ) ∈ ( C ∗ ) m , t ∈ C ∗ ・ When | a k | < 1 ( k = 1 , . . . , m ), | t | < 1, a standard choice for the n -cycle C n is the real torus T n { } R = | z 1 | = · · · = | z n | = 1 . When the parameters go out from this domain, the n -cycle should be deformed accordingly.