Non-Gaussian integrals and general hypergeometric functions
aa r X i v : . [ m a t h . G M ] S e p NON-GAUSSIAN INTEGRALS AND GENERALHYPERGEOMETRIC FUNCTIONS
ALEXANDER ROI STOYANOVSKY
Abstract.
By a non-Gaussian integral we mean integral of theproduct of an arbitrary function and exponent of a polynomial.We develop a theory of such integrals, which generalizes and sim-plifies the theory of general hypergeometric functions in the senseof I. M. Gelfand et al.
Introduction
Let(1) P ( t , . . . , t n ) = X ω =( ω ,...,ω n ) c ω t ω , t ω = t ω . . . t ω n n , be a polynomial in n variables with complex coefficients, and let α ( t , . . . , t n ) be an arbitrary (possibly multi-valued) function. By the non-Gaussian integral transform of α or simply by the non-Gaussian inte-gral we mean the integral(2) I α ( P ) = I α ( c ω ) = I e P ( t ,...,t n ) α ( t , . . . , t n ) dt . . . dt n , considered as a function of the coefficients c ω of the polynomial P . Herethe symbol H means that integration goes over a real n -dimensional ori-ented contour without boundary in the n -dimensional complex space,equipped with a choice of a continuous branch of the integrated func-tion. Since the integration contour can be non-unique, integral (2) is,in general, a multi-valued function.In the last decades it became increasingly clear that theory of non-Gaussian integrals plays a fundamental role in mathematics and phys-ics. Some insights to this theory were given in [1, 2, 7–12], but the maincontribution is due to I. M. Gelfand et al. [3–5]. They have shown thattheory of non-Gaussian integrals is closely related with the theory ofgeneral hypergeometric functions invented by I. M. Gelfand et al.In the present paper we generalize and simplify the theory of generalhypergeometric functions. Namely, we study general integrals of the form(3) I β ( P , . . . , P k ) = I β ( P , . . . , P k , t , . . . , t n ) dt . . . dt n , where P , . . . , P k are polynomials in t , . . . , t n , and β ( y , . . . , y k , t , . . . , t n ) is an arbitrary (possibly multi-valued) function. We call theseintegrals, considered as functions of coefficients of the polynomials P , . . . , P k , by ( general ) hypergeometric functions . They are expressedthrough non-Gaussian integral (2) as(4) I β ( P , . . . , P k ) = I α ( P ) , where(5) P ( λ , . . . , λ k , t , . . . , t n ) = λ P + . . . + λ k P k ,λ , . . . , λ k are additional variables (the Cayley trick ), and α ( λ , . . . , λ k , t , . . . , t n ) is the inverse Fourier–Laplace transform of β ( y , . . . , y k , t , . . . , t n ) with respect to y , . . . , y k . The theory of these integrals is a gen-eralization and simplification of the theory of general hypergeometricfunctions due to Gelfand et al. This generalization is similar to pass-ing from toric algebraic geometry to general algebraic geometry. Theonly more general point in the theory of Gelfand et al. is that in theirtheory the polynomials P and P , . . . , P k can be Laurent polynomials,i. e. polynomials in t j and t − j .Let us say a few words about foundations of our theory. In thepaper we use, without formalization, the notion “arbitrary function” (ofseveral variables). The general intuition of function goes back to Eulerand, for more modern times, to I. M. Gelfand. Informally, a function isa dependence of a variable on several variables, in particular, given bycompositions of operations + , − , × , :, derivative, integral, and solvingequations. The most close set theory formalizations of this notion arethe notions of distribution [6], multi-valued analytical function, D -module, and sheaf.The paper is organized as follows. In § § § A -hypergeometric systems and GG -systems defined andstudied in [3–5].Finally, in § Acknowledgment.
I would like to thank A. Gemintern for interestto this work.
ON-GAUSSIAN INTEGRALS AND HYPERGEOMETRIC FUNCTIONS 3 Examples of non-Gaussian integrals
Recall that by the non-Gaussian integral we mean integral (2).
Example P = c t + . . . + c n t n be a linear form in t , . . . , t n ,then(6) I α ( c t + . . . + c n t n ) = b α ( c , . . . , c n )is the Fourier–Laplace transform of α ( t , . . . , t n ). Example P be a quadratic expression in t , . . . , t n , thenintegral (2) is well studied and called the Gaussian integral transformof α or simply by the Gaussian integral . For description of the imageof the Gaussian integral transform, see [9, 11] (the results of [9] areannounced in [10]).
Example α ( t , . . . , t n ) ≡
1, then we obtain what we call the proper non-Gaussian integral, which we denote simply by(7) I ( P ) = I α ≡ ( P ) = I e P ( t ,...,t n ) dt . . . dt n . Arbitrary integral (2) is expressed through I ( P ) as(8) I α ( P ) = 1(2 π ) n I I ( P − µ t − . . . − µ n t n ) b α ( µ , . . . , µ n ) dµ . . . dµ n , where b α ( µ , . . . , µ n ) is the Fourier–Laplace transform of α ( t , . . . , t n ).Function I ( P − µ t − . . . − µ n t n ) for fixed P was considered in [2]. Example α ( t , . . . , t n ) = t u − . . . t u n − n , where u , . . . , u n arecomplex numbers. Then integral (2) has been studied in [3] and calleda GG -function . We denote it by(9) I t u − ...t un − n ( P ) = GG ( P ; u , . . . , u n ) = GG ( c ω ; u , . . . , u n ) . Arbitrary integral (2) is expressed through the GG -function as(10) I α ( P ) = I GG ( P ; u , . . . , u n ) e α ( u , . . . , u n ) du . . . du n , where e α ( u , . . . , u n ) is the inverse Mellin transform of α ( t , . . . , t n ) t . . . t n . Example P , . . . , P k be polynomials in t , . . . , t n , let λ , . . . , λ k be additional variables, let P ( λ , . . . , λ k , t , . . . , t n ) be defined by theCayley trick (5), and let α ( λ , . . . , λ k , t , . . . , t n ) be a function. Then ALEXANDER ROI STOYANOVSKY we have(11) I α ( P ) = I e λ P + ... + λ k P k α ( λ , . . . , λ k , t , . . . , t n ) dλ . . . dλ k dt . . . dt n = I β ( P ( t , . . . , t n ) , . . . , P k ( t , . . . , t n ) , t , . . . , t n ) dt . . . dt n , where β ( y , . . . , y k , t , . . . , t n ) is the Fourier–Laplace transform of α ( λ , . . . , λ k , t , . . . , t n ) with respect to λ , . . . , λ k . Definition.
We call integral (11) by a ( general ) hypergeometricfunction of coefficients of the polynomials P , . . . , P k , and denote it by I β ( P , . . . , P k ).2. Examples of general hypergeometric functions
Recall that by a general hypergeometric function we mean integral(3) or (11).
Example k = 1 and β ( y, t , . . . , t n ) = e y γ ( t , . . . , t n ), where γ ( t , . . . , t n ) is any function, then we obtain non-Gaussian integral (2)(with γ instead of α ). Example y , . . . , y k be real numbers, let(12) β ( y , . . . , y k , t , . . . , t n ) = θ ( y − y ) . . . θ ( y k − y k ) γ ( t , . . . , t n ) , where γ ( t , . . . , t n ) is any function,(13) θ ( y ) = 0 if y < θ ( y ) = 1 if y ≥ , and let P , . . . , P k be polynomials with real coefficients. Then we obtainthat the integral of γ ( t , . . . , t n ) over the semi-algebraic domain(14) P ( t , . . . , t n ) ≤ y , . . . , P k ( t , . . . , t n ) ≤ y k in the n -dimensional real space is a hypergeometric function of coeffi-cients of the polynomials P , . . . , P k . Example the generalized Sturmfels theorem. (Cf. [13, 12]) Let k = 1 and(15) β ( y, t , . . . , t n ) = − πi log( y − y ) ∂γ∂t ( t , . . . , t n ) , ON-GAUSSIAN INTEGRALS AND HYPERGEOMETRIC FUNCTIONS 5 where γ ( t , . . . , t n ) is any function. Integrating by parts, we obtainthat the integral(16) − πi I log( P ( t , . . . , t n ) − y ) ∂γ∂t ( t , . . . , t n ) dt . . . dt n = 12 πi I ∂P/∂t P ( t , . . . , t n ) − y γ ( t , . . . , t n ) dt . . . dt n = I P ( t ,...,t n )= y γ ( t , . . . , t n ) dt . . . dt n is a hypergeometric function of coefficients of the polynomial P . Inparticular, if n = 1, then we obtain the following theorem. Theorem 2.1.
For a root x of a polynomial equation P ( t ) = y andfor any function γ ( t ) , the quantity γ ( x ) is a ( multi-valued ) hypergeo-metric function of coefficients of the polynomial P .Example k = 1 and(17) β ( y, t , . . . , t n ) = δ ( y − y ) γ ( t , . . . , t n ) or β ( y, t , . . . , t n ) = 12 πi ( y − y ) γ ( t , . . . , t n ) , where γ ( t , . . . , t n ) is any function. Then we obtain that the Gelfand–Leray integral [6](18) I P ( t ,...,t n )= y γ ( t , . . . , t n ) dt . . . dt n /dP is a hypergeometric function of coefficients of the polynomial P . An-other proof of this fact follows from the generalized Sturmfels theorem:it suffices to differentiate equality (16) with respect to y and replace ∂γ/∂t with γ .In particular, if n = 1, then we obtain the following theorem. Theorem 2.2.
For a root x of a polynomial equation P ( t ) = y and for any function γ ( t ) , the quantity γ ( x ) /P ′ ( x ) is a ( multi-valued ) hypergeometric function of coefficients of the polynomial P .Example β ( y , . . . , y k , t , . . . , t n ) = y v . . . y v k k t u − . . . t u n − n . Then we obtain that the generalized Euler integral (20) I P ( t , . . . , t n ) v . . . P k ( t , . . . , t n ) v k t u − . . . t u n − n dt . . . dt n is a hypergeometric function of coefficients of the polynomials P , . . . , P k . ALEXANDER ROI STOYANOVSKY
Example l ≤ k and(21) β ( y , . . . , y k , t , . . . , t n ) = δ ( y ) . . . δ ( y l ) γ ( y l +1 , . . . , y k , t , . . . , t n ) , where γ ( y l +1 , . . . , y k , t , . . . , t n ) is any function. Then we obtain thatthe Gelfand–Leray integral [6](22) I P = ... = P l =0 γ ( P l +1 , . . . , P k , t , . . . , t n ) dt . . . dt n /dP . . . dP l is a hypergeometric function of coefficients of the polynomials P , . . . , P k .In particular, if l = n , then we obtain that for a solution ( x , . . . , x n )of the system of equations(23) P ( x , . . . , x n ) = . . . = P n ( x , . . . , x n ) = 0 , the quantity(24) γ ( P n +1 ( x , . . . , x n ) , . . . , P k ( x , . . . , x n ) , x , . . . , x n ) /J ( x , . . . , x n )is a hypergeometric function of coefficients of the polynomials P , . . . , P k ,where(25) J ( x , . . . , x n ) = det( ∂P i /∂t j ) ≤ i,j ≤ n ( x , . . . , x n )is the Jacobian of the polynomials P , . . . , P n at the point t = x , . . . , t n = x n .3. Equations satisfied by non-Gaussian integrals and bygeneral hypergeometric functions
Equations satisfied by the non-Gaussian integral. Propo-sition 3.1.
Non-Gaussian integral (2) satisfies the following system ofequations :(26) ∂I α ∂c ω ( P ) = I t ω α ( P ) for any ω , provided that the integral is regular in c ω , i. e. admitsdifferentiation under the sign of integral ;(27) I ∂α∂tj ( P ) = − I α ∂P∂tj ( P ) , j = 1 , . . . , n. Corollary.
Non-Gaussian integral (2) satisfies the equations (28) ∂I α ∂c ω = ∂ ω ∂c ω . . . ∂ ω n ∂c ω n n I α ON-GAUSSIAN INTEGRALS AND HYPERGEOMETRIC FUNCTIONS 7 for any ω , where c j is the coefficient before the linear monomial t j in P , j = 1 , . . . , n ;(29) I t j ∂α∂tj = − I α − X ω ω j c ω ∂I α ∂c ω , j = 1 , . . . , n ;(30) ∂∂c ω . . . ∂∂c ω N I α = ∂∂c ω ′ . . . ∂∂c ω ′ N ′ I α for any N, N ′ and any ω , . . . , ω N , ω ′ , . . . , ω ′ N ′ such that (31) ω + . . . + ω N = ω ′ + . . . + ω ′ N ′ . System (26, 29) almost coincides with the GG -system from [3]. Sys-tem (29, 30) almost coincides with a corollary of the GG -system calledin [3] by the A -system with variables c ω , ω ∈ A , where A is a finite setof exponents ω = ( ω , . . . , ω n ) ∈ Z n of monomials t ω = t ω . . . t ω n n . The A -system consists of equations (30) and the equations(32) X ω ω j c ω ∂I α ∂c ω = − u j I α , j = 1 , . . . , n, where u j are complex numbers (parameters). The GG -system and the A -system are satisfied by the GG -function (9).3.2. Equations satisfied by the general hypergeometric func-tion. Proposition 3.2.
General hypergeometric integral (3, 11) , con-sidered as a function of coefficients c ( i ) ω of polynomials P i , i = 1 , . . . , k ,satisfies the following system of equations :(33) ∂I β ∂c ( i ) ω ( P , . . . , P k ) = I t ω ∂β∂yi ( P , . . . , P k ) for any ω and i , provided that the integral is regular in c ( i ) ω , i. e. admitsdifferentiation under the sign of integral ;(34) I y i β ( P , . . . , P k ) = I P i β ( P , . . . , P k ) , i = 1 , . . . , k ;(35) I ∂β∂tj ( P , . . . , P k ) = − k X i =1 I ∂β∂yi ∂Pi∂tj ( P , . . . , P k ) , j = 1 , . . . , n. Corollary.
General hypergeometric integral satisfies the equations (36) ∂∂c ( i )0 ! ω + ... + ω n − ∂∂c ( i ) ω I β = ∂∂c ( i )1 ! ω . . . (cid:18) ∂∂c ( i ) n (cid:19) ω n I β ALEXANDER ROI STOYANOVSKY for any ω and i , where c ( i )0 is the constant term of P i and c ( i ) j is the co-efficient before the linear monomial t j in P i , i = 1 , . . . , k , j = 1 , . . . , n ;(37) I y i ∂β∂yi = X ω c ( i ) ω ∂I β ∂c ( i ) ω , i = 1 , . . . , k ;(38) I t j ∂β∂tj = − I β − X ω,i ω j c ( i ) ω ∂I β ∂c ( i ) ω , j = 1 , . . . , n ;(39) ∂∂c ( i ) ω . . . ∂∂c ( i N ) ω N I β = ∂∂c ( i ) ω ′ . . . ∂∂c ( i N ) ω ′ N I β for any N , any i , . . . , i N and any ω , . . . , ω N , ω ′ , . . . , ω ′ N such that (40) ω + . . . + ω N = ω ′ + . . . + ω ′ N . System (37–39) almost coincides with the e A -hypergeometric system( e A -system) from [3–5] with variables c ( i ) ω , ω ∈ A i , i = 1 , . . . , k , where A i is a finite set of exponents ω ∈ Z n , and the set e A ⊂ Z n + k = Z n × Z k is defined as(41) e A = A × { e } ∪ . . . ∪ A k × { e k } , where e , . . . , e k is the standard basis in Z k (the Cayley trick [5]). The e A -system consists of equations (39) and the equations(42) X ω c ( i ) ω ∂I β ∂c ( i ) ω = v i I β , i = 1 , . . . , k ;(43) X ω,i ω j c ( i ) ω ∂I β ∂c ( i ) ω = − u j I β , j = 1 , . . . , n, where v i , u j are complex numbers (parameters). The e A -system is sat-isfied by the generalized Euler integral (20).4. Power series expansions
Power series expansions of non-Gaussian integrals.
Let(44) P ( t , . . . , t n ) = X ω ∈ A c ω t ω , (45) P ( t , . . . , t n ) = X ω ∈ A c ω t ω = P ( t , . . . , t n ) + X ω ∈ A a ω t ω , ON-GAUSSIAN INTEGRALS AND HYPERGEOMETRIC FUNCTIONS 9 where A ⊂ Z n is a finite set. Assume that non-Gaussian integral I α ( P )(2) is regular in a neighborhood of P .Following [3], let us call a set of exponents B = { ω , . . . , ω n } ⊂ A a base if they are linearly independent, i. e. if they form a basis in C n .We shall give the expansion of I α ( P ) = I α ( c ω ) ω ∈ A into a power seriesin the variables(46) a ω = c ω − c ω , ω ∈ A \ B, with coefficients being functions of a j = a ω j = c ω j − c ω j , j = 1 , . . . , n .To this end, let us make the change of variables(47) T j = t ω j , j = 1 , . . . , n, in integral (2). We obtain(48) I α ( P ) = I e n P j =1 a j t ωj + P ω ∈ A \ B a ω t ω α ( t , . . . , t n ) dt . . . dt n = I e n P j =1 a j T j + P ω ∈ A \ B a ω T lω α ( T , . . . , T n ) dT . . . dT n , where(49) α ( t , . . . , t n ) = e P ( t ,...,t n ) α ( t , . . . , t n ) , (50) α ( T , . . . , T n ) dT . . . dT n = α ( t , . . . , t n ) dt . . . dt n , and l ω = ( l ω , . . . , l nω ) is the vector of coordinates of ω with respect tothe basis ω , . . . , ω n ,(51) n X j =1 l jω ω j = ω. The numbers l jω are, in general, rational numbers.Expanding (48) into a power series in a ω , we obtain(52) I α ( P ) = X m ω ≥ ω ∈ A \ B C m ( a , . . . , a n ) Y ω a m ω ω m ω ! , where(53) C m ( a , . . . , a n ) = I e n P j =1 a j T j T P ω ∈ A \ B m ω l ω α ( T , . . . , T n ) dT . . . dT n is the Fourier–Laplace transform of T P ω ∈ A \ B m ω l ω α ( T , . . . , T n ).In particular, if α ( t , . . . , t n ) = t u − . . . t u n − n and P = 0, then weobtain the expansion of GG -function (9) into a power series of hyper-geometric type [3]. If B is the standard basis in Z n , then we obtain the obvious expansionof I α ( P ) into a power series in a ω , ω ∈ A ,(54) I α ( P ) = X m ω ≥ ω ∈ A Y ω a m ω ω m ω ! I t P mωω α ( P ) . Power series expansions of general hypergeometric func-tions.