Hypergeometric generating functions for values of Dirichlet and other L functions

Abstract

Although there is vast literature on the values of L functions at nonpositive integers, the recent appearance of some of these values as the coefficients of specializations of knot invariants comes as a surprise. Using work of G. E. Andrews [(1981) Adv. Math. 41, 173–185; (1986) q-Series: Their Development and Application in Analysis, Combinatories, Physics, and Computer Algebra, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics 66 (Am. Math. Soc, Providence, RI); (1975) Problems and Prospects for Basic Hypergeometric Series: The Theory and Application of Special Functions (Academic, New York); and (1992) Illinois J. Math. 36, 251–274], we revisit this old subject and provide uniform and general results giving such generating functions as specializations of basic hypergeometric functions. For example, we obtain such generating functions for all nontrivial Dirichlet L functions.

1. Introduction and Statement of Results

Suppose that χ₁₂ is the Dirichlet character with modulus 12 defined by

and let be its associated L function. In a recent paper, Zagier (1) obtained the following generating function for the values of L(s, χ₁₂) at negative odd integers

[1.1]

The t-series expansion on the left is obtained by using the Taylor series expansion for e^–t. In other recent works (see refs. 2–4; K. Hikami, personal communication), similar generating functions have been obtained for the values of certain L functions at nonpositive integers. These examples were derived from q-series identities associated to the “summation of the tails” of a modular form and the combinatorics of q-difference equations.

These L-function values are generalized Bernoulli numbers (for example, see ref. 5), and there is vast literature on the subject. These numbers play many important roles in number theory. However, their recent appearance in knot theory, algebraic geometry, and mathematical physics comes as a surprise. In the recent works of Hikami (4) and Zagier (1), they appear as the coefficients of specializations of Kashaev invariants in knot theory.

Motivated by these examples, it is natural to seek general results that uniformly provide such L values as the coefficients of generating functions like Eq. 1.1 and the multidimensional series appearing in ref. 4. Here we provide such results.

If 0 < c ≤ 1 is a rational number, then let ζ(s, c), for (s) > 1, denote the Hurwitz zeta function

[1.2]

These functions possess an analytic continuation to with the exception of a simple pole at s = 1 with residue 1. These zeta functions, which generalize Riemann's zeta function, are the building blocks of many important L functions. All of the L functions in refs. 1 and 4 are finite linear combinations of functions of the form

[1.3]

where 0 < b < a are integers. We obtain multidimensional basic hypergeometric generating functions for all of these L functions at nonpositive integers. (For a survey of basic hypergeometric series, see refs. 6 or 7.)

We shall use the standard notation

[1.4]

and throughout we assume that <1 and that the other parameters are restricted to domains that do not contain any singularities of the series or products under consideration. For integers k ≥ 2, define the series F_k(z; q) by

[1.5]

Theorem 1. If 0 < b < a are integers and k ≥ 2, then

The d/dz series in Theorem 1 is obtained by differentiating summand by summand in z, and then setting z = 1 and q = e^–t. Although we omit its closed form expression for brevity, we note that it is easily obtained from Eq. 1.5 by using the standard rules for differentiation.

Suppose that χ mod f is a nontrivial Dirichlet character (i.e. a homomorphism from ) with Dirichlet L function

[1.6]

Such a χ is even (respectively odd) if χ(–1) = 1 [respectively χ(–1) = –1]. We give basic hypergeometric generating functions for the values of L(s, χ) at nonpositive integers. Define the series G(z; q) by

[1.7]

If χ mod f is a Dirichlet character, then define G_χ(z; q) by

[1.8]

Theorem 2. Suppose that χ mod f is a nontrivial Dirichlet character.

1. If χ is odd, then

2. If χ is even, then

Suppose that χ is a nontrivial even (respectively odd) Dirichlet character. It is a classical fact that if n ≥ 0 is even (respectively odd), then L(–n, χ) = 0. Therefore, Theorem 2 provides the L values at nonpositive integers for all Dirichlet L functions.

In section 2, we give formulas for F_k(z; q) and G(z; q). To prove these identities, we use important results of Andrews (9–12). In section 3 we recall classical facts about Mellin integral representations of L functions, and then combine them with the identities of section 2 to prove Theorems 1 and 2. In section 4 we give several examples of Theorems 1 and 2.

2. q-Series Identities

For convenience, we use the following abbreviation:

[2.1]

Theorem 2.1. If k ≥ 2, then the following identity is true

Theorem 2.2. The following identity is true

Remarks. The series F₂(z; q) is a specialization of the classical Rogers–Fine identity (8)

whereas the series G(z; q) is a specialization of an identity of Andrews (9) related to “false” theta functions in Ramanujan's lost notebook.

Proofs of Theorem 2.1 and 2.2. Two sequences (α_n, β_n) are said to form a Bailey pair with respect to a if for every n ≥ 0,

Given such a pair, Andrews (10) has shown that for any natural number k and complex numbers b_i, c_i we have (subject to convergence conditions)

[2.2]

The theorems follow by inserting the right Bailey pairs into the above equation and making appropriate specializations of the parameters. Substituting the Bailey pair with respect to a (9),

yields a limiting case of Andrews' multidimensional generalization of Watson's transformation (11),

[2.3]

Keeping in mind that and (y/x)_r → 1 as x → ∞, let a = z², b₁ =–z, c_k = z, b_k = q, and all remaining b_i, c_i → ∞ in Eq. 2.3. After a bit of simplification, this is Theorem 2.1.

Next insert in Eq. 2.2 the Bailey pair with respect to q (12),

The k = 1 case is Andrews' generalized false theta identity (9),

Letting b = q, z = zq, and c → ∞ gives Theorem 2.2.

3. Proofs of Theorems 1 and 2

To prove Theorems 1 and 2, we require the following classical results regarding the Mellin integral representations of L functions.

Proposition 3.1. Suppose that ψ is a periodic function with modulus f with mean value zero, and let

As t ↘ 0 we have

Proof. By the hypothesis on ψ, L(s) has an analytic continuation to . Suppose that H(t) is the asymptotic expansion as t ↘ 0 given by

[3.1]

Using the Mellin integral representation for L(2s) (for (s) > 1), and by letting T = n²t, it turns out that

[3.2]

For any N > 0, this combined with Eq. 3.1 implies that

where F(s) is analytic for (s) > –N. Therefore, b(n) is the residue at s =–>n of Γ(s)L(2s), and so

The same argument applies to the asymptotic expansion of We now prove Theorems 1 and 2 by using Theorems 2.1 and 2.2 and Proposition 3.1. Proof of Theorem 1. By Theorem 2.1, we have

[3.3]

By letting z = 1 and q = e^–t in Eq. 3.3, we obtain a power series in t. To see that the t series is well defined, we use the fact that the constant term in the Taylor expansion of 1 – e^–mt is zero for every positive integer m. Each summand in Eq. 1.5 therefore has the property that the numerator contains 2n_k–1 such factors whereas the denominator has n_k–1 many [note: the (–z; q)_n1+1 does not contribute any]. Therefore, Theorem 1 for L_a,b(–2n) follows from Proposition 3.1.

To obtain Theorem 1 for L_a,b(–2n – 1), we apply the argument above to the series that is obtained by differentiating Eq. 3.3 in z summand by summand before letting z = 1 and q = e^–t. As above, the form of Eq. 1.5 implies that the resulting t series is well defined.

Proof of Theorem 2. By Theorem 2.2, if 0 < r < f, then

[3.4]

Recall the definition of G_χ(z; q) (see Eq. 1.8)

[3.5]

By Eq. 3.4, this implies that

Therefore, we have

An inspection of Eq. 1.7 shows that both t series are well defined. Since χ(f/2) = 0 for even f, Proposition 3.1 implies that if χ is odd, then

Similarly if χ is even, then Proposition 3.1 implies that

This completes the proof of Theorem 2.

4. Examples

Here we give examples of Theorems 1 and 2. Example 1. Here we give the k = 2 example of Theorem 1 for the function

where χ_–1 is the unique nontrivial Dirichlet character modulo 4. Theorem 1 implies that

Theorem 1 provides a generating function for the values L(–2n – 1, χ_–1), where n ≥ 0. Since χ_–1 is odd, these values are zero. Theorem 1 implies that

Example 2. Here we compute the values of L(s, χ_–1) at even nonpositive integers again by using Theorem 2. By Theorem 2, we find that

Example 3. Suppose that χ₅ is the Dirichlet character modulo 5 given by the Legendre symbol modulo 5. By definition, we have that

and so we have

By differentiating in z summand by summand, then letting z = 1 and q = e^–t, Theorem 2 gives