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A Lauricella Hypergeometric Series Over Finite Fields 2 A LAURICELLA HYPERGEOMETRIC SERIES OVER FINITE FIELDS BING HE Abstract. In this paper we introduce a finite field analogue for a Lauricella hypergeometric series. An integral formula for the Lauricella hypergeometric series and its finite field analogue are deduced. Transformation and reduction formulae and several generating functions for the Lauricella hypergeometric series over finite fields are obtained. Some of these generalize certain results of Li et al and Greene as well as several other known results. Contents 1. Introduction 1 2. Another expression 4 3. An integral formula and its finite field analogue 6 4. Reduction and Transformation formulae 11 5. Generating functions 18 Acknowledgement 21 References 21 1. Introduction F F∗ Let q be a power of a prime and let q and q denote the finite field of q elements and F∗ the group of multiplicative characters of q respectively. Setting χ(0) = 0 for all characters, F∗ F we extend the domain of all characters χ of q bto q. Let χ and ε denote the inverse of χ arXiv:1610.04473v3 [math.CA] 1 May 2017 and the trivial character respectively. See [3] and [9, Chapter 8] for more information about characters. Following [2], we define the generalized hypergeometric function as ∞ a0, a1, . , an (a0)k(a1)k · · · (an)k k n+1Fn x := x , b1, . , bn k!(b1)k · · · (bn)k k=0 X where (z)k is the Pochhammer symbol given by (z)0 = 1, (z)k = z(z + 1) · · · (z + k − 1) for k ≥ 1. 2010 Mathematics Subject Classification. Primary 33C65, 11T24; Secondary 11L05, 33C20. Key words and phrases. Lauricella hypergeometric series over finite fields, reduction formula, transformation for- mula, generating function. 1 A LAURICELLA HYPERGEOMETRIC SERIES OVER FINITE FIELDS 2 It was Greene who in [8] developed the theory of hypergeometric functions over finite fields and established numerous transformation and summation identities for hypergeometric se- ries over finite fields which are analogues to those in the classical case. Greene, in particular, introduced the notation A, B G BC(−1) F x = ε(x) B(y)BC(1 − y)A(1 − xy) 2 1 C q y X for A,B,C ∈ Fq and x ∈ Fq, that is a finite field analogue for the integral representation of Gauss hypergeometric series [2]: b a, b Γ(c) 1 dt F x = tb(1 − t)c−b(1 − tx)−a , 2 1 c Γ(b)Γ(c − b) ˆ t(1 − t) 0 and defined the finite field analogue for the binomial coefficient as A G B(−1) = J(A, B), B q where J(χ, λ) is the Jacobi sum given by J(χ, λ)= χ(u)λ(1 − u). u X For more details about the finite field analogue for the generalized hypergeometric functions, please see [6, 7, 12]. In this paper, for the sake of simplicity, we define the finite field analogue for the binomial coefficient and the classic Gauss hypergeometric series by A A G = q = B(−1)J(A, B). B B and A, B A, B G F x = q · F x = ε(x)BC(−1) B(y)BC(1 − y)A(1 − xy), 2 1 C 2 1 C y X respectively. There are many interesting double hypergeometric functions in the field of hypergeometric functions. Among these functions, the Appell series F1 may be one of the most important functions: ′ ′ (a)m+n(b)m(b )n m n F1(a; b, b ; c; x,y)= x y , |x| < 1, |y| < 1. m!n!(c)m n m,n≥ + X0 See [2, 5, 14] for more material about the Appell series. Inspired by Greene’s work, Li et al [11] gave a finite field analogue for the Appell series F1 and established some transformation and reduction formulas and the generating functions for the function over finite fields. In that paper, the finite field analogue for the Appell series F1 was given by ′ ′ F1(A; B,B ; C; x,y)= ε(xy)AC(−1) A(u)AC(1 − u)B(1 − ux)B (1 − uy). u X A LAURICELLA HYPERGEOMETRIC SERIES OVER FINITE FIELDS 3 It was Lauricella [10] who in 1893 generalized the Appell series F1 to the Lauricella (n) hypergeometric series FD that is defined by ∞ ∞ (n) a; b1, · · · , bn (a)m1+···+mn (b1)m1 · · · (bn)mn m1 mn FD x1, · · · ,xn := · · · x1 · · · xn . c (c)m1+···+mn m1! · · · mn! m1=0 mn=0 X X It is clear that ′ (2) a; b, b b, a (1) a; b F (a; b, b′; c; x,y)= F x,y and F x = F x . 1 D c 2 1 c D c (n) so the Lauricella hypergeometric series F D is an n-variable extension of the Appell series F1 and the hypergeometric function 2F1. Motivated by the work of Greene [8] and Li et al [11], we give a finite field analogue for (n) the Lauricella hypergeometric series. Since the Lauricella hypergeometric series FD has a integral representation 1 a−1 c−a−1 (n) a; b1, · · · , bn Γ(c) u (1 − u) F x1, · · · ,xn = du, D c Γ(a)Γ(c − a) ˆ (1 − x u)b1 · · · (1 − x u)bn 0 1 n we give the finite field analogue for the Lauricella hypergeometric series in the following form: n A; B , · · · ,B F ( ) 1 n x , · · · ,x D C 1 n = ε(x · · · xn)AC (−1) A(u)AC(1 − u)B (1 − x u) · · · Bn(1 − xnu), 1 1 1 u X where A, B1, · · · ,Bn,C1, · · · ,Cn ∈ Fq, x1, · · · ,xn ∈ Fq and the sum ranges over all the F Γ(c) elements of q. In the above definition, the factor Γ(a)Γ(c−a) is dropped to obtain simpler b results. We choose the factor ε(x1 · · · xn)AC(−1) to get a better expression in terms of binomial coefficients. From the definition of the Lauricella hypergeometric series over finite fields, we know that A; B,B′ A, B B; A F (A; B,B′; C; x,y)= F (2) x,y and F x = F (1) x . 1 D C 2 1 C D C Then the Lauricella hypergeometric series over finite fields can be regarded as an n -variable extension of the finite field analogues for the Appell series F1 and the hypergeometric function 2F1. From Theorem 2.1 or the definition, we know that the Lauricella hypergeometric series over finite fields n A; B , · · · ,B F ( ) 1 n x , · · · ,x D C 1 n is invariant under permutation of the subscripts 1, 2, · · · , n, namely, it is invariant under permutation of the B′s and x′s together. The aim of this paper is to give several transformation and reduction formulas and the generating functions for the Lauricella hypergeometric series over finite fields. We know that the Lauricella hypergeometric series over finite fields is an n-variable extension of the A LAURICELLA HYPERGEOMETRIC SERIES OVER FINITE FIELDS 4 finite field analogues for the Appell series F1 and the hypergeometric function 2F1. So some of the results in this paper are generalizations of certain results in [8, 11] and some other known results. For example, [11, Theorem 1.3] and [8, Theorem 3.6] are special cases of Theorem 2.1. We will give another expression for the Lauricella hypergeometric series over finite fields in the next section. An integral formula for the Lauricella hypergeometric series and its finite field analogue are deduced in Section 3. In Section 4, several transformation and reduction formulae for the Lauricella hypergeometric series over finite fields will be given. The last section is devoted to some generating functions for the Lauricella hypergeometric series over finite fields. 2. Another expression In this section we give another expression for the Lauricella hypergeometric series over finite fields. Theorem 2.1. For A, B1, · · · ,Bn,C ∈ Fq and x1, · · · ,xn ∈ Fq, we have n A; B , · · · ,B F ( ) 1 n x , · · · ,x b D C 1 n 1 Aχ1 · · · χn B1χ1 Bnχn = n · · · χ1(x1) · · · χn(xn), (q − 1) Cχ · · · χn χ χn χ1,··· ,χ 1 1 X n where each sum ranges over all multiplicative characters of Fq. To carry out our study, we need some auxiliary results which will be used in the sequel. The results in the following proposition follows readily from some properties of Jacobi sums. Proposition 2.1. (See [8, (2.6), (2.8) and (2.12)]) If A, B ∈ Fq, then A A (2.1) = , b B AB A B (2.2) = AB(−1), B A A A (2.3) = = −1 + (q − 1)δ(A), ε A where δ(χ) is a function on characters given by 1 if χ = ε δ(χ)= . 0 otherwise The following result is a finite field analogue for the well-known identity a c c c − b = . b a b a − b A LAURICELLA HYPERGEOMETRIC SERIES OVER FINITE FIELDS 5 Proposition 2.2. (See [8, (2.15)])For A,B,C ∈ Fq, we have A C C CB = − (q − 1)B(−1)b δ(A) + (q − 1)AB(−1)δ(BC). B A B AB The following result is also very important in the derivation of Theorem 2.1. Theorem 2.2. (Binomial theorem over finite fields, see [8, (2.10)]) For A ∈ Fq and x ∈ Fq, we have 1 Aχ b A(1 − x)= δ(x)+ χ(x), q χ − 1 χ X where the sum ranges over all multiplicative characters of Fq and δ(x) is a function on Fq given by 1 if x = 0 δ(x)= . 0 if x 6= 0 We are now ready to prove Theorem 2.1.
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