A -SERIES IDENTITY AND THE ARITHMETIC OF HURWITZ ZETA FUNCTIONS GWYNNETH H COOGAN AND KEN ONO Introduction and Statement of Results In a recent paper [?], D Zagier used a -series identity to prove that e t/24 e t ) e 2t ) e nt ) Lχ 2, 2n ) t/24)n 2 where χ 2 is the Kronecker character for Q 2) Subseuently, the second author, G E Andrews and J Jimene Urro [?] obtained two infinite families of such -series identities In ad hoc cases, these identities produced further generating functions for L-function and eta values such as e t/8 e 2t ) e 4t ) e 2nt ) + e t ) + e 3t ) + e 2n+)t ) 2 Lχ 2, 2n ) t/8)n Here χ 2 is the Kronecker character for Q 2) Since these results are born out of the arithmetic of basic hypergeometric functions for more on basic hypergeometric functions, see [?]), we refer to them as hypergeometric Clearly, this method for computing ζ-values at non-positive integral arguments is very different from the classical approach involving special values of Bernoulli polynomials Here we show that such hypergeometric generating functions are widespread, and may be obtained in a systematic, rather than ad hoc, fashion We begin by recalling the Jacobi Triple Product Identity 2n ) + 2 2n ) + 2 2n ) 2n n2 n n Our results depend on a hypergeometric analog of this classical result We begin by fixing notation Let a; ) :, and for every positive integer n let a; ) n : a) a) a 2 ) a n ) ) 99 Mathematics Subject Classification B65, M35 The authors thank the National Science Foundation for their generous support The second author also thanks the Alfred P Sloan Foundation and the David and Lucile Packard Foundation for their generous support
Proposition The following formal power series identity is true: A; ) : + ; ) n n ) n 2n n2 ; ) n This identity follows from an identity due to Rogers and Fine [?] We include a proof in the next section for the benefit of those unfamiliar with basic hypergeometric series The following is the result of a simple calculation Corollary 2 If b < a and ζ and ζ are complex numbers, then Θa, b, ζ, ζ ;, ) : Z b Q b2 AZ a/2 Q ab ; Q a2 ) ) n ζ an+b ζ an+b)2 an+b) an+b)2, where Z : ζ and Q : ζ To illustrate our results regarding hypergeometric generating functions for ζ-values, consider the following series Θ2,,, ;, ) : + 2 ; 4 ) n n 2n ) n 2n+) 2n+)2 2) 2 6 ; 4 ) n Since the Taylor series of e nt is e nt t n t2n 2 + t3n 6, we find that Θ2,,, ;, e t ) is a well defined formal power series in t Θ2,,, ;, e t ) e t + e e 2t ; e 4t ) n e 2nt 2t e 6t ; e 4t ) n 2 + 2 t + 5 4 t2 + 3) Observe that the naive substitution of and e t in the right hand side of 2) does not produce a well defined t-series It turns out that the series in??) satisfies Θ2,,, ;, e t ) 2 + 2 t + Lχ, 2n) t)n 4) Here χ is the non-trivial Dirichlet character with modulus 4 Therefore, 4) is a hypergeometric generating function whose coefficients interpolate values of Lχ, s) at non-positive even integers If < c is a rational number, then let ζs, c) denote the Hurwit eta-function ζs, c) : n + c) s 5) 2
These functions possess an analytic continuation to the whole complex plane, with the exception of a simple pole at s It is easy to see that 4) is euivalent to Θ2,,, ;, e t ) ζ 2n, /4) ζ 2n, 3/4)) 6t)n 6) The preceding example illustrates our first general result Theorem 3 Suppose that b < a are integers, and suppose that α, β and m are positive integers for which both ) ord 2 m) < ord 2 2a 2 β), 2) aα + 2aβb + ak) mod 2m) for some residue class k mod 2m) If ζ : e 2πiα/m and ζ : e 2πiβ/m, then as a formal power series in t we have Θa, b, ζ, ζ ;, e t ) m c j ζ 2n, 2aj + b ) d j ζ j where c j : ζ 2ja+b) ζ 2ja+b)2 2n,, and d j : ζ 2j+)a+b) ζ 2j+)a+b)2 ) ) 2j + )a + b 4a 2 m 2 t) n, Here we highlight one special case of Theorem 3 The following corollary provides the generating function for the values at non-positive even integers for certain differences of Hurwit eta functions Corollary 4 If b < a, then as a formal power series in t we have ) Θa, b,, ;, e t b ) ζ 2n, ζ 2n, a + b )) 4a 2 t) n 2a 2a Note that??) is a special case of this result We now obtain further such series using Corollary 2 To state these results, let Θ a, b, ζ, ζ ;, ) denote the series obtained by differentiating, summand by summand, Θa, b, ζ, ζ ;, ) with respect to Theorem 5 Assuming the notation and hypotheses in Theorem 3, as a formal power series in t we have Θ a, b, ζ, ζ ;, e t ) m c j ζ j where c j : ζ 2aj+b 2n, 2ja + b ) d j ζ 2n, ζ 2aj+b)2, and d j : ζ 2j+)a+b) ζ 2j+)a+b)2 As a corollary, we have the following counterpart to Corollary 4 3 )) 2j + )a + b 4a 2 m 2 t) n,
4 Corollary 6 If b < a, then as a formal power series in t we have Θ a, b,, ;, e t ) ) b ζ 2n, ζ 2n, a + b )) 4a 2 t) n 2a 2a 2a 2 Proof of Proposition We prove Proposition following the method described in [?] Proof of Proposition Consider the function ; ) n t n B, t; ) : B n t n, 2) ; ) n Note that A; ) B, ; ) We determine its transformation under the map + B, t; ) B, t; ), and iterate it to obtain the proposition The fact that ; ) n ); ) n implies B, t; ) + ) tb, t; ) 22) + ) Since B n+ + n+ ) B n n ), by multiplying by t n+ and summing on n we obtain B n+ t n+ + B n+ t) n+ t B n t n t B n t) n This implies the following transformation formula B, t; ) + ) t) Transformations 22) and 23) imply B, t; ) t) t) One iteration of this transformation formula yields B, t; ) t) t) + t) B, t; ) 23) t) ) + t) tb, t; ) t) + ) ) + t) t2 ) t) + ) t) t + ) ) + t) + t) t) t) + ) + 2 ) 2 t 2 4 B 2, t 2 ; ) An infinite number of iterations yields B, t; ) t ; ) n t; ) n t 2n ) ) n n t n n2 ; ) n t; ) n
5 If t above, the cancellation among the -factorial terms above yields A; ) + B, ; ) ) n 2n n2 3 Proof of Theorems 3 and 5 Proof of Theorems 3 and 5 We let and replace by e t t + t 2 /2 +, and notice that the sum is a convergent power series in t when ζ a/2 ) vanishes ζ ab+ka2 for infinitely many integers k and + ζ a/2 ζ ab+ca2 ) does not vanish for any integer c To guarantee that the numerators of the summands in this series start with larger and larger powers of t, we reuire that ζ a/2 This implies that ζ ab+ka2 for infinitely many integers k aα + 2βab + ka 2 ) mod 2m) 34) This is the second hypothesis in Theorem 3 Next, if both the numerator and the denominator vanish, then for some k and c k + d Z ζ a/2 ζ ab ζ a2 k ζ a2 k + ζ a2 c and + ζ a/2 ζ ab ζ a2 c e 2πia 2 kβ m + e 2πi a2 dβ 2a 2 dβ m ) m 2n +, for d and some odd integer 2n + However, this is prohibited by the first hypothesis in Theorem 3 Therefore, the formal power series in t is well defined Define coefficients cn) by Θa, b, ζ, ζ ;, e t ) cn)t n, 35) ) n ζ an+b ζ an+b)2 We consider the Mellin transform of Θa, b, ζ, ζ ;, e t ) : Θa, b, ζ, ζ ;, e t )t s dt ) n ζ an+b ) n ζ an+b ) n ζ an+b an + b) 2s e tan+b)2 ζ an+b)2 ζ an+b)2 ζ an+b)2 e tan+b)2 t s dt e tan+b)2 t s dt e t t s dt Γs)Lψ, 2s) 36)
Here Lψ, s) is the generalied L-function defined by analytically continuing the Dirichlet series ψn) Lψ, s) :, n s where ψn) : ) n b)/a ζ n ζ n2 if n b mod a), and otherwise Since ψ has period we can write Lψ, s) as a linear combination of Hurwit eta functions which turns out as m Lψ, s) ) s n Lψ, s) ) s ψr)ζ j r s, c j ζ s, 2aj + b ) d j ζ s, r ) 6 )) a2j + ) + b, 37) where c j ζ 2aj+b ζ 2aj+b)2, and d j ζ a2j+)+b ζ a2j+)+b)2 On the other hand if we are careful about the way in which we evalute the integral in the Mellin transform, we are able to locate and identify its residues Θa, b, ζ, ζ ;, e t )t s dt + N N Θa, b, ζ, ζ ;, e t )t s dt Θa, b, ζ, ζ ;, e t )t s dt cn)t n+s dt + cn)t n+s + Ot N+s )dt + cn) s + n + As) Θa, b, ζ, ζ ;, e t )t s dt Θa, b, ζ, ζ ;, e t )t s dt where because e t is small for large t, As) is analytic for Res) > N It is clear now that the residue at s n is cn) By 36), this implies that cn) Res s n Lψ, 2s)Γs)) )n Lψ, 2n) Therefore we find that Θa, b, ζ, ζ ;, e t ) n Lψ, 2n)t n )
Theorem 3 now follows from??) To prove Theorem 5, observe that the product rule implies that Θ a, b, ζ, ζ ;, e t ) converges under the same hypotheses in Theorem 3 The only difference in the calculation of the t expansion for the derivative is the factor an + b) which ends up yielding Γs)Lψ, 2s ) in the analog of 36) The rest of the proof of Theorem 5 follows mutatis mutandis References [] GE Andrews, J Jimene-Urro, K Ono, -series identities and values of certain L functions, Duke Math J, 8, 2, pages 395-49 [2] N Fine, Basic hypergeometric series and applications, Math Surv and Mono 27, Amer Math Soc 988 [3] D Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology, to appear Department of Mathematics, University of Wisconsin, Madison, Wisconsin 5376 E-mail address: gwynneth@mathwiscedu Department of Mathematics, University of Wisconsin, Madison, Wisconsin 5376 E-mail address: ono@mathwiscedu 7