A Product Formula for the Twisted Lefschetz Zeta Function

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A New Class of Infinite Products Generalizing Viète's Product Formula for π

We show how functions F(z) which satisfy an identity of the form F(α z) = g(F(z)) for some complex number α and some function g(z) give rise to infinite product formulas that generalize Viète's product formula for π. Specifically, using elliptic and trigonometric functions we derive closed form expressions for some of these infinite products. By evaluating the expressions at certain points we obtain formulas expressing infinite products involving nested radicals in terms of well-known constants. In particular, simple infinite products for π and the lemniscate constant are obtained. 2000 Mathematics Subject Classification: Primary—40A20; Secondary—33E05     

A Formula for the Partition Function That “Counts”

We derive a combinatorial multisum expression for the number D(n, k) of partitions of n with Durfee square of order k. An immediate corollary is therefore a combinatorial formula for p(n), the number of partitions of n. We then study D(n, k) as a quasipolynomial. We consider the natural polynomial approximation \({\tilde{D}(n, k)}\) to the quasipolynomial representation of D(n, k). Numerically, the sum \({\sum_{1\leq k \leq \sqrt{n}} \tilde{D}(n, k)}\) appears to be extremely close to the initial term of the Hardy-Ramanujan-Rademacher convergent series for p(n).     

Remarks on the Trotter–Kato Product Formula for Unitary Groups

Let A and B be non-negative self-adjoint operators in a separable Hilbert space such that their form sum C is densely defined. It is shown that the Trotter product formula holds for imaginary parameter values in the L ²-norm, that is, one has

$ \lim_{n\to+\infty} \int\limits^T_{-T} \left\|\left(e^{-itA/n}e^{-itB/n} \right)^nh - e^{-itC}h\right\|^2dt = 0 $

for each element h of the Hilbert space and any T > 0. This result is extended to the class of holomorphic Kato functions, to which the exponential function belongs. Moreover, for a class of admissible functions: \({\phi(\cdot),\psi(\cdot):{\mathbb R}_+ \longrightarrow {\mathbb C}}\), where \({{\mathbb R}_+ := [0,\infty)}\), satisfying in addition \({{\Re{\rm e}}\,(\phi(y))\ge 0, {\Im{\rm m}}\,(\phi(y) \le 0}\) and \({{\Im{\rm m}}\,(\psi(y)) \le 0}\) for \({y \in {\mathbb R}_+}\), we prove that

$ \,\mbox{\rm s-}\hspace{-2pt} \lim_{n\to\infty}(\phi(tA/n)\psi(tB/n))^n = e^{-itC} $

holds true uniformly on \({[0,T]\ni t}\) for any T > 0.

     

An Explicit Formula for Szegő Kernels on the Heisenberg Group

In this paper, we give an explicit formula for the Szegö kernel for (0, q) forms on the Heisenberg group H_n+1.     

An Explicit Formula for the Coefficients in Laplace’s Method

Laplace’s method is one of the fundamental techniques in the asymptotic approximation of integrals. The coefficients appearing in the resulting asymptotic expansion arise as the coefficients of a convergent or asymptotic series of a function defined in an implicit form. Due to the tedious computation of these coefficients, most standard textbooks on asymptotic approximations of integrals do not give explicit formulas for them. Nevertheless, we can find some more or less explicit representations for the coefficients in the literature: Perron’s formula gives them in terms of derivatives of an explicit function; Campbell, Fröman and Walles simplified Perron’s method by computing these derivatives using an explicit recurrence relation. The most recent contribution is due to Wojdylo, who rediscovered the Campbell, Fröman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials. In this paper, we provide an alternative representation for the coefficients that contains ordinary potential polynomials. The proof is based on Perron’s formula and a theorem of Comtet. The asymptotic expansions of the gamma function and the incomplete gamma function are given as illustrations.     

Generalized Lindelöf Finiteness Conditions for the λ-Type of a Subharmonic Function

We establish a finiteness criterion for the -type of a subharmonic function. In the case where (r) = r L(r), , where L is a slowly varying function, this criterion coincides with the Lindelöf criterion.     

A Construction of the Rational Function Sheaves on Elliptic Curves

Abstract The authors introduce an effective method to construct the rational function sheaf κ on an elliptic curve E, and further study the relationship between κ and any coherent sheaf on E. Finally, it is shown that the category of all coherent sheaves of finite length on E is completely characterized by κ.     

An inequality for the function π(n)

We prove that the inequality $\pi ^2 \left( m \right) + \pi ^2 \left( n \right) \leqslant \tfrac{5} {4}\pi ^2 \left( {m + n} \right)$ holds for all integers m, n ≥ 2. The constant factor 5/4 is sharp. This complements a result of Panaitopol, who showed in 2001 that ½ π ²(m+ n) ≤ π ²(m) + π ²(n) is valid for all m, n ≥ 2. Here, as usual, π(n) denotes the number of primes not exceeding n.     

An Extension of the Rényi Formula

In this paper, as a natural extension of the Rényi formula which counts labeled connected unicyclic graphs, we present a formula for the number of labeled(k + 1)-uniform(p, q)-unicycles as follows:U(k+1)p, q={p!/2[(k-1)!]~q·∑qt=2(q~(q-t-1)· sgn(tk- 2))/(q- t)!, p = qk,0, p≠qk,where k, p, q are positive integers.     

On the Waring Problem for Infinite Matrices

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An Approximate Functional Equation for the Primitive of Hardy’s Function

A formula of Atkinson type for the primitive of Hardy’s function is generalized to the case where the lengths of the two sums involved in that formula vary in wide ranges.     

Explicit Formula for the Hyperbolic Scattering Determinant

We prove the explicit formula for the hyperbolic scattering determinant in the case of a general subgroup F of PSL （2, R）. The class of test functions involved （not necessarily odd nor continuous） is much broader than that previously known. As an application of the technique, a new representation of the Millson-Shintani zeta function is obtained.     

An Explicit Derivation of the Möbius Function for Bruhat Order

We give an explicit nonrecursive complete matching for the Hasse diagram of the strong Bruhat order of any interval in any Coxeter group. This yields a new derivation of the Möbius function, recovering a classical result due to Verma.     

Green’s Function for Second Order Elliptic Equations in Non-divergence Form

We construct the Green function for second order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition and the domain has C^1,1 boundary. We also obtain pointwise bounds for the Green functions and its derivatives.

     

On the Linear Complexity of the Naor–Reingold Pseudo-random Function from Elliptic Curves

We show that the elliptic curve analogue of the pseudo-random number function, introduced recently by M. Naor and O. Reingold, produces a sequence with large linear complexity. This result generalizes a similar result of F. Griffin and I. E. Shparlinski for the linear complexity of the original function of M. Naor and O. Reingold. The proof is based on some results about the distribution of subset-products in finite fields and some properties of division polynomials of elliptic curves.     

An Extension of Itô's Formula for Anticipating Processes

In this paper we introduce a class of square integrable processes, denoted by L^F, defined in the canonical probability space of the Brownian motion, which contains both the adapted processes and the processes in the Sobolev space L^2,2. The processes in the class L^F satisfy that for any time t, they are twice weakly differentiable in the sense of the stochastic calculus of variations in points (r, s) such that r s t. On the other hand, processes belonging to the class L^F are Skorohod integrable, and the indefinite Skorohod integral has properties similar to those of the Ito integral. In particular we prove a change-of-variable formula that extends the classical Itô formula. Those results are generalization of similar properties proved by Nualart and Pardoux⁽⁷⁾ for processes in L^2,2.     

On the Chebyshev Function ψ(x)

Suppose that the Riemann hypothesis is valid, and let be the Chebyshev function. In this paper, we obtain the following bound for all >1 and positive integers .     

The Mehler Formula for the Generalized Clifford-Hermite Polynomials

The Mehler formula for the Hermite polynomials allows for an integral representation of the one-dimensional Fractional Fourier transform. In this paper, we introduce a multi-dimensional Fractional Fourier transform in the framework of Clifford analysis. By showing that it coincides with the classical tensorial approach we are able to prove Mehler＇s formula for the generalized Clifford-Hermite polynomials of Clifford analysis.     

On the Multiple Solutions of Dirichlet Problems for semilinear Elliptic Equations (Critical Increasing Case)

In this paper we study Dirichlet problem: