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Mellin's Generalized Hypergeometric Functions Appendix A Mellin’s Generalized Hypergeometric Functions Here, we will introduce a generalized hypergeometric series generalizing the ν! appearing in the denominator of the coefficient of hypergeometric series (3.1) § n defined in 3.1 to i=1 bi(ν)!, where bi(ν)’s are integer-valued linear forms on the lattice L = Zn, and will derive a system of partial differential equations and integral representation of Euler type. We will also explain its relation to the Bernstein−Mellin−Sato b-function. We use the notation defined in § 3.1. A.1 Definition An element a of the dual lattice L∨ of the lattice L = Zn naturally extends to a linear form on L⊗C. Here and after, we denote this linear form on C by the same symbol a.WedenoteapointofL as ν,apointofL⊗C as s,and if necessary, we denote them by a(ν), a(s) to distinguish them. + ∈ ∨ ≤ ≤ Suppose that m + n integer-valued linear forms aj L ,1 j m, − ∈ ∨ ≤ ≤ ai L ,1 i n on L and m α =(α1, ··· ,αm) ∈ C are given satisfying the following two assumptions: − det(ai (ek)) =0, (A.1) − ··· − ∨ ⊗ C (namely, a1 (s), ,an (s)regardedaselementsofL are linearly inde- pendent) and the condition corresponding to (3.2) m n + − ≤ ≤ aj (ek)= ai (ek), 1 k n. (A.2) j=1 i=1 Then, we consider the following Laurent series K. Aomoto et al., Theory of Hypergeometric Functions, Springer Monographs 261 in Mathematics, DOI 10.1007/978-4-431-53938-4, c Springer 2011 262 A Mellin’s Generalized Hypergeometric Functions m + Γ (a (ν)+αj) F (α, x)= j=1 j xν (A.3) n Γ (a−(ν)+1) ν∈C∩L i=1 i as a function of x ∈ Cn, which is called Mellin’s hypergeometric series of general type. Here, C is the cone in L ⊗ R = Rn C { ∈ Rn| − ≥ ≤ ≤ } := s ai (s) 0, 1 i n (A.4) − ··· − defined by linearly independent elements a1 (s), ,an (s)aselementsof L∨ ⊗ R, and the sum is taken over all of the lattice points in C.Moreover, for each coefficient of (A.3) to be finite, we impose the condition: for any ν ∈C∩L, + ∈ Z ≤ ≤ aj (ν)+αj / ≤0, 1 j m. (A.5) Fig. A.1 Integrals related to Mellin’s hypergeometric series have been treated in many references, for example in [Ad-Sp], [A-K], [Ao9], [Bai], [Bel], [G-G-R], [Kim], [Kit2], [Ok], [Sl]. A.2 Kummer’s Method Following § 3.3, we would like to show the convergence of this series and ele- mentary integral representation of Euler type at the same time by Kummer’s method. For this purpose, we first prepare three lemmata. A.2 Kummer’s Method 263 % % m + n − By j=1 aj (ν)= i=1 ai (ν) which follows easily from (A.2), using (3.18), we obtain the following lemma: Lemma A.1. Set m−1 m−1 U =(u1, ··· ,um−1) ∈ C ,um :=1− uj, j=1 m + − aj (ν)+αj 1 U(u):= uj ,ω= dU/U, j=1 Δm−1(ω): the twisted cycle defined from the (m − 1)-simplex m m−1 m−1 Δ = {u ∈ R |uj ≥ 0, 1 ≤ j ≤ m}, |α| = αj. j=1 Under the assumption αj ∈ C \ Z, 1 ≤ j ≤ m, we have m Γ (a+(ν)+α ) j%=1 j j ··· m + = U(u)du1 dum−1. (A.6) | | − Γ ( j=1 aj (ν)+ α ) Δm 1(ω) Next, for later use, we generalize the multinomial theorem which played an essential role in the proof of Theorem 3.2. For this purpose, we first prepare the following fact about the zero-dimensional toric varieties (see [Od]fortoric varieties). Lemma A.2. For a given y ∈ (C∗)n, we denote the set of finite points defined by the equations (a zero-dimensional toric variety) by P (y): − ∗ a (e ) a−(e ) ∈ C n| 1 i ··· n i ≤ ≤ P (y):= w ( ) w1 wn = yi, 1 i n . ∈ − ≤ ≤ ∈ Then, if there exists s L such that νi = ai (s), 1 i n for a ν L,we have wν = |P (y)|ys, (A.7) w∈P (y) and for a ν ∈ L that does not admit such expression, we have wν =0. (A.8) w∈P (y) Here, |P (y)| signifies the cardinality of P (y). Proof. Since any of yi,1≤ i ≤ n are not zero, we can define its argument and we fix one among them, and we denote by log yi the branch of logarithms 264 A Mellin’s Generalized Hypergeometric Functions defined by it. A condition to be w ∈ P (y) can be expressed by using a solution of √ − ··· − − a1 (ei)X1 + + an (ei)Xn =logyi +2riπ 1, 1 ≤ i ≤ n, ri ∈ Z, − Xi ≤ ≤ as wi = e ,1 i n. By the condition (A.1)andthefactthatai (ek) − ∈ Z \{ } is an integer, we have√ det(ai (ek)) 0 and the number of such ··· − X1, ,Xn mod 2π 1 is finite by the% Cramer formula. Hence,% P (y)isa ν − finite set. Next, notice that w =exp( νkXk)=Πi exp(si k ak (ei)Xk). When each si is an integer, this concludes (A.7). When ν cannot be ex- − ∈ pressed as νi = ai (s), s L, by the above consideration, at least one of si’s is a rational number which is not an integer, from which we have r−1 √ e2π −1k/r =0, k=0 and (A.8) follows. A.3 Toric Multinomial Theorem As the third lemma, we state a fact that should be called a toric version of the ordinary multinomial theorem. Lemma A.3. Set −|α| (1 − w1 ···−wn) Ψ| |(y):= . α |P (y)| w∈P (y) If any w ∈ P (y) satisfies |w1| + ···+ |wn| < 1, then the multinomial expansion % n − | | Γ ( i=1 ai (s)+ α ) s Ψ| |(y)= y (A.9) α Γ (|α|) n Γ (a−(s)+1) s∈C∩L i=1 i holds and it converges uniformly on every compact subset. − ··· − − %Proof. Here, we abbreviate (a1 (s), ,an (s)) as a (s). Under the condition n | | i=1 wi < 1, the series arising in the following process converge uniformly on every compact subset: A.3 Toric Multinomial Theorem 265 ) *−|α| n (|α|; |ν|) 1 − w /|P (y)| = wν /|P (y)| i ν! ∈ ∈ ∈Zn w P (y) i=1 w P (y) ν ≥0 (|α|; |ν|) wν = ν! |P (y)| ∈Zn ∈ ν ≥0 w P (y) (|α|; |ν|) = ys (by (A.7)and(A.8)) ∈Zn ν! ν ≥0 ν=a−(s) s∈L % Γ ( a−(s)+|α|) = i ys (by (A.4)). Γ (|α|) n Γ (a−(s)+1) s∈C∩L i=1 i Remark A.1. The sum in the right-hand side of (A.9) is taken over the lattice points of C∩L, and as we see from Figure A.1, in general, negative integers appear as components of s ∈C∩L. Hence, the (A.9) converges not on an ap- propriate neighborhood of the% origin as in the ordinary multinomial theorem C∗ n { n | | } but on the image of ( ) ∩ i =1 w1 < 1 by − − a (e ) a (e ) 1 i ··· n i ≤ ≤ yi = wi wn , 1 i n. (A.10) To show the convergence of the hypergeometric series (A.3), we state a little bit on the convergent domain of a Laurent series in several variables. To simplify the notation, we set ∗ n ηi exp Γ :={y ∈ (C ) ||yi| = e , (η1, ··· ,ηn) ∈ Γ }, for a domain Γ in Rn. Then, setting 1 Γ := ζ ∈ Rn| ζ < log , 1 ≤ i ≤ n , ζ i 2n we clearly have n ∗ n exp Γζ ⊂ (C ) ∩ |wi| < 1 . i=1 Now, we consider the map induced from (A.10): 266 A Mellin’s Generalized Hypergeometric Functions Φ : Rn −→ Rn, ζ −→ η, − ······ − ηi = a1 (ei)ζ1 + + an (ei)ζn, ζi =log|wi|,ηi =log|yi|, 1 ≤ i ≤ n. By Assumption (A.1), Φ is a non-singular linear map and Φ(Γζ )=Γη be- comes a infinite convex domain surrounded by n hyperplanes. Clearly, (A.9) ∗ n converges absolutely and uniformly on exp Γη ⊂ (C ) . Fig. A.2 A.4 Elementary Integral Representations Now, we have prepared to apply Kummer’s method. The rest of the argu- ments can be formally developed as follows. First, rewriting (A.3), we have % − + Γ ( a (ν)+|α|) Γ (a (ν)+αj) F (α, x)= i xν % j Γ (a−(ν)+1) Γ ( a−(ν)+|α|) ν∈C∩L i i % − | | Γ ( ai (ν)+ α ) ν = − x U(u)du1 ···dum−1 Γ (a (ν)+1) Δm−1(ω) ν∈C∩L i (by (A.6)) % m − − Γ ( a (ν)+|α|) αj 1 · i ν ··· = uj − y du1 dum−1. Δm−1(ω) Γ (a (ν)+1) ν∈C∩L j=1 i Here, we set A.4 Elementary Integral Representations 267 m a+(e ) j i ≤ ≤ yi = xi uj , 1 i n. j=1 By the construction of the twisted cycle Δm−1(ω)(cf.§ 3.2), independent of + ∈ m−1 the sign of aj (ei), for u Δ (ω), there exists positive constants c1 <c2 such that m + aj (ei) c1 ≤ u ≤ c2, 1 ≤ i ≤ n. j j=1 Hence, we have m + aj (ei) log |yi| =log|xi| +log u , 1 ≤ i ≤ n, j j=1 and as Γη is an infinite domain surrounded by n hyperplanes, there exists an n appropriate vector c ∈ R such that if we always have (log |x1|, ··· , log |xn|) ∈ m−1 c + Γη for any u ∈ Δ (ω), then we can let (log |y1|, ··· , log |yn|) ∈ Γη.
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