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Given a formal map $F=(F_1...,F_n)$ of the form $z+\text{higher}$ order terms, we give tree expansion formulas and associated algorithms for the D-Log of F and the formal flow F_t. The coefficients which appear in these formulas can be viewed as certain generalizations of the Bernoulli numbers and the Bernoulli polynomials. Moreover the coefficient polynomials in the formal flow formula coincide with the strict order polynomials in combinatorics for the partially ordered sets induced by trees. Applications of these formulas to the Jacobian Conjecture are discussed., Comment: Latex, 32 pages

We determine an expression 9 (1X) for the virtual Euler characteristics of the moduli spaces of s-pointed real (-y = 1/2) and complex (-y = 1) algebraic curves. In particular, for the space of real curves of genus g with a fixed point free involution, we find that the Euler characteristic is (-2)s-1(1-29-1)(g?s-2)!Bg/g! where Bg is the gth Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is (-1) S (g+s-2)!Bg+ /(g+1) (g -1) ! The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to counit cells. The approach involves a parameter y that permits specialization of the formula to the real and complex cases. This suggests that 4 (-y) itself may describe the Euler characteristics of some related moduli spaces, although we do not yet know what these spaces might be.

Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields. In this paper, we shall improve Shintani's formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [4] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise. 1. IDENTITIES AMONG BERNOULLI NUMBERS The Bernoulli numbers Bn (n = 0,1, 2, ..) are defined by t _00 Bntn et1 E n! ' Itl 1 since the function t t et _1 2 is an even function of t by direct verification. Bernoulli numbers are used to express the special values of Riemann zeta function

From the perspective of the well-known identity \[ 2 F 1 ( a , b ; c ; 1 ) = Γ ( c ) Γ ( c − a − b ) Γ ( c − a ) Γ ( c − b ) , {}_2{F_1}(a,b;c;1) = \frac {{\Gamma (c)\Gamma (c - a - b)}} {{\Gamma (c - a)\Gamma (c - b)}}, \] we clarify the connections between the Stirling numbers s k n s_k^n and the Riemann zeta function ζ ( n ) \zeta (n) . As a consequence, certain series and integrals can be evaluated in terms of ζ ( n ) \zeta (n) and s k n s_k^n .

We study the zeros ( mod p ) \pmod p of the polynomial β p ( X ) = Σ k ( B k / k ) ( X p − 1 − k − 1 ) {\beta _p}(X) = {\Sigma _k}({B_k}/k)({X^{p - 1 - k}} - 1) for p p an odd prime, where B k {B_k} denotes the k k th Bernoulli number and the summation extends over 1 ⩽ k ⩽ p − 2 1 \leqslant k \leqslant p - 2 . We establish a reciprocity law which relates the congruence β p ( r ) ≡ 0 ( mod p ) {\beta _p}(r) \equiv 0\;\pmod p to a congruence f p ( n ) ≡ 0 ( mod r ) {f_p}(n) \equiv 0\,\pmod r for r r a prime less than p p and n ∈ Z n \in {\mathbf {Z}} . The polynomial f p ( x ) {f_p}(x) is the irreducible polynomial over Q {\mathbf {Q}} of the number Tr L Q ( ζ ) ⁡ ζ \operatorname {Tr}_L^{{\mathbf {Q}}(\zeta )}\zeta , where ζ \zeta is a primitive p 2 {p^2} th root of unity and L ⊂ Q ( ζ ) L \subset {\mathbf {Q}}(\zeta ) is the extension of degree p p over Q {\mathbf {Q}} . These congruences are closely related to the prime divisors of the indices I ( α ) = ( O : Z [ α ] ) I(\alpha ) = (\mathcal {O}:{\mathbf {Z}}[\alpha ]) , where O \mathcal {O} is the integral closure in L L and α ∈ O \alpha \in \mathcal {O} is of degree p p over Q {\mathbf {Q}} . We establish congruences ( mod p ) \pmod p involving the numbers I ( α ) I(\alpha ) and show that their prime divisors r ≠ p r \ne p are closely related to the congruence r p − 1 ≡ 1 ( mod p 2 ) {r^{p - 1}} \equiv 1\,\pmod {p^2} .

A formula due to Euler and Legendre is used to construct finite-difference counterparts to the Dirichlet space. The spaces have integral representations and characterizations in terms of area integrals. Their reproducing kernels are logarithms of the reproducing kernels of the Newton spaces, which are counterparts to the Hardy class. A Hilbert space with reproducing kernel \[ log ⁡ [ ( 1 / w ¯ z ) log 1 / ( 1 − w ¯ z ) ] \log [(1/\overline w z)\,\log \;1/(1 - \overline w z)] \] is also shown to exist and to be related to Bernoulli numbers and combinatorial theory.

A regularized version of J. Diamond’sp-adic log gamma function and hisp-adic Euler constants are represented as integrals using B. Mazur’sp-adic Bernoulli measure.

We prove general forms of von Staudt’s theorems on the Bernoulli numbers. As a consequence we are able to deduce strong versions of a number of congruences involving various generalisations of the Bernoulli numbers. For example we obtain an improved form of a congruence due to Hurwitz involving the Laurent series coefficients of the Weierstrass elliptic function associated with a square lattice.

We consider twelve infinite systems of polynomials in z which for z = 1 degenerate either to the numbers of Bernoulli or Euler, or to others simply dependent upon these. The first part proceeds from the definition of anharmonic polynomials to the specific twelve systems discussed; the second presents an adaptation of the symbolic calculus of Blissard and Lucas in sufficient detail for rapidly developing a simple isomorphism between the algebra of the polynomials and that of the twelve elliptic functions sn, cn, ns, nc, sc, ... of Glaisher, and the third contains a short selection from the simpler algebraic and congruential relations between the polynomials. Incidentally there is pointed out in the second part a new interpretation of Kronecker's work on certain symmetric functions and their connections with Bernoulli's nlumbers. Owing to the length of the paper the development stops short of the quadratic transformation of the polynomials which corresponds to the transformation of the second order in elliptic functions, but the material given is a necessary foundation for all higher transformations. For the same reason only prime moduli are considered in the congruences, although the case in which the modulus is a power of a prime can be treated in essentially the same way, but at greater length. All references are at the end of the paper.