Asymptotic expansions for the distribution of quadratic forms in normal variables

Higher order asymptotic expansions for the distribution of quadratic forms in normal variables are obtained. The Cornish-Fisher inverse expansions for the percentiles of the distribution are also given. The resulting formula for a definite quadratic form guarantees accuracy almost up to fourth decimal place if the distribution is not very skew. The normalizing transformation investigated by Jensen and Solomon (1972, J. Amer. Statist. Assoc., 67, 898–902) is reconsidered based on the rate of convergence to the normal distribution.Faculty of Science, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812 Japan     

On the distribution and moments of the fourier coefficients of cusp forms

Two theorems about the distribution and moments of the Fourier coefficients of cusp forms are proved in this paper. Project supported by the National Natural Science Foundation of China     

On the coefficients of integrated expansions of Bessel polynomials

A new formula expressing explicitly the integrals of Bessel polynomials of any degree and for any order in terms of the Bessel polynomials themselves is proved. Another new explicit formula relating the Bessel coefficients of an expansion for infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is also established. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.     

On product expansions of generalized modular forms

We give examples of generalized modular forms with the property that their divisors are supported at the cusps and the exponents in their q-product expansions take infinitely many values.      

Asymptotic expansions of the binomial coefficients

A general formulae for the asymptotic expansion of not centered binomial coefficients are derived and some useful estimates of the binomial coefficients are presented. The sum of the binomial coefficients is also studied.     

Asymptotic expansions for the coefficients of analytic generating functions

PairsF(x), G(x) of analytic generating functions that satisfy relations such as 1+G(x)=exp(F(x)) are studied. It is shown that, ifF(x) satisfies fairly mild regularity conditions, such as those imposed by Hayman in his study of coefficients of some general classes of functions, thenG(x) satisfies the much stricter conditions imposed by Harris and Schoenfeld. This enables one to obtain complete asymptotic expansions for the coefficients ofG(x). Applications of this result are made to enumerations of trees.Dedicated to Professor Janos Aczél on his 60th birthday     

On normal characterizations by the distribution of linear forms,assuming finite variance

If X₁ and X₂ are independent and identically distributed (i. i. d.) with finite variance, then $\text{(X}{}_1\text{+X}{}_2\text{)}\text{√2}$ has the same distribution as X₁ if and only if X₁ is normal with mean zero (Pólya [9]). The idea of using linear combinations of i. i. d. random variables to characterize the normal has since been extended to the case where σ^∞_i=1a_iX_i has the same distribution as X₁. In particular if at least two of the a_i's are non-zero and X₁ has finite variance, then Laha and Lukacs [8] showed that X₁ is normal. They also [7] established the same result without the assumption of finite variance. The purpose of this note is to present a different and easier proof of the characterization under the assumption of finite variance. The idea of the proof follows closely the approach used by Pólya in [9]. The same technique is also used to give a characterization of the exponential distribution.     

Congruences for modular forms and generalized Frobenius partitions

The Ramanujan Journal - The partition function is known to exhibit beautiful congruences that are often proved using the theory of modular forms. In this paper, we study the extent to which these...     

On the stability of characterization of the normal distribution by the properties of linear forms

If the assumptions of the characterization theorem are fulfilled not exactly but only approximately, then can we state that the conclusion of this characterization is also fulfilled approximately? Theorems in which this kind of problems are considered are called stability theorems or the stability of characterizations. The problems of the stability of characterization of the normal distribution by means of identically distributed linear form and monomial are considered in a uniform metric. The moment and symmetry conditions are considerably weaker in comparison to previously published versions of the theorem. The choice of a uniform metric appears to be very natural in the framework of Zolotarev’s methodology advising to investigate stability.     

On the Fourier coefficients of certain Hilbert modular forms

We prove that given any \(\epsilon >0\), a non-zero adelic Hilbert cusp form \({\mathbf {f}}\) of weight \(k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n\) and square-free level \(\mathfrak {n}\) with Fourier coefficients \(C_{{\mathbf {f}}}(\mathfrak {m})\), there exists a square-free integral ideal \(\mathfrak {m}\) with \(N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }\) such that \(C_{{\mathbf {f}}}(\mathfrak {m})\ne 0\). The implied constant depends on \(\epsilon , F\).

     

On a characterization of the normal distribution by means of identically distributed linear forms

Let X₁, X₂,…, be independent, identically distributed random variables. Suppose that the linear forms L₁ = Σ_j=1^∞a_jX_j and L₂ = Σ_j=1^∞b_jX_j exist with probability one and are identically distributed; necessary and sufficient conditions assuring that X₁ is normally distributed are presented. The result is an extension of a theorem of Linnik (Ukrainian Math. J.5 (1953), 207–243, 247–290) concerning the case that the linear forms L₁ and L₂ have a finite number of nonvanishing components. This proof only makes use of elementary properties of characteristic functions and of meromorphic functions.     

On the signs of Fourier coefficients of Hilbert cusp forms

The Ramanujan Journal - We prove that given any $$\epsilon > 0$$ and a primitive adelic Hilbert cusp form f of weight $$k=(k_1,k_2,\ldots ,k_n) \in (2 {\mathbb {Z}})^n$$ and full level,...     

On the Frobenius manifolds for cusp singularities

We show that the Frobenius manifold associated to the pair of a cusp singularity and its canonical primitive form is isomorphic to the one constructed from the Gromov–Witten theory for an orbifold projective line with at most three orbifold points. We also calculate the intersection form of the Frobenius manifold.     

On the Frobenius problem for Beatty sequences

We study the solvability of linear diophantine equations in two variables within the context of Beatty sequences.