Related papers: Stirling's Series Revisited
We calculate full asymptotic expansions of prime-independent multiplicative functions on additive arithmetic semigroups that satisfy a strong form of Knopfmacher's axioms. When applied to the semigroup of unlabeled graphs, our method yields…
We consider the spherical integral of real symmetric or Hermitian matrices when the rank of one matrix is one. We prove the existence of the full asymptotic expansions of these spherical integrals and derive the first and the second term in…
In this paper, we consider a generalization of the Stirling number sequence of both kinds by using a specialization of a new family of symmetric functions. We give combinatorial interpretations for this symmetric functions by means of…
This paper presents a family of rapidly convergent summation formulas for various finite sums of analytic functions. These summation formulas are obtained by applying a series acceleration transformation involving Stirling numbers of the…
Let $F$ be an entire function of exponential type represented by the Taylor series \[ F(z) = \sum_{n\ge 0} \omega_n \frac{z^n}{n!} \] with unimodular coefficients $|\omega_n|=1$. We show that either the counting function $n_F(r)$ of zeroes…
We construct a new type of convergent, and asymptotic, representations, dyadic expansions. Their convergence is geometric and the region of convergence often extends from infinity down to $0^+$. We show that dyadic expansions are…
Defining a family of recurrences, we generalize Comtet's formula for the generating function of the enumeration of indecomposable permutations. Consequently, we generalize Panaitopol's asymptotic expansion for the prime counting function,…
We give asymptotic expressions for the number of commuting matrices over finite fields. For this, we use product expansions for the corresponding generating functions.
Asymptotic expansions are derived for solutions of the parabolic cylinder and Weber differential equations. In addition the inhomogeneous versions of the equations are considered, for the case of polynomial forcing terms. The expansions…
In this paper, the logarithmically complete monotonicity property for a functions involving $q$-gamma function is investigated for $q\in(0,1).$ As applications of this results, some new inequalities for the $q$-gamma function are…
Stirling approximation of the factorials and multinominal coefficients are generalized based on the one-parameter ($\kappa$) deformed functions introduced by Kaniadakis [Phys. Rev. E \textbf{66} (2002) 056125]. We have obtained the relation…
This article derives full asymptotic expansions for integrals of the form \[ \int_{0}^{1}f(u)(1+q\cdot u^{n})^{w/n}du \] as $n\rightarrow\infty$, with parameters real $w\neq 0$ and $q\in(-1,1]$, or positive $w$ for $q=-1$. We relate the…
We give estimates for the convolution product of an arbitrary number of endlessly continuable functions. This allows us to deal with nonlinear operations for the corresponding resurgent series, e.g. substitution into a convergent power…
In [8], asymptotic expansion of the martingale with mixed normal limit was provided. The expansion formula is expressed by the adjoint of a random symbol with coefficients described by the Malliavin calculus, differently from the standard…
A difficult and long-standing problem in mathematical physics concerns the determination of the value of $f(\infty)$ from the asymptotic series for $f(x)$ about $x\!=\!0$. In the past the approach has been to convert the asymptotic series…
Some problems in the theory and applications of stochastic processes can be reduced to solving integral equations. While explicit solutions for these equations are often elusive, valuable insights can be gained through their asymptotic…
Asymptotic expansion is presented for an estimator of the Hurst coefficient of a fractional Brownian motion. For this, a recently developed theory of asymptotic expansion of the distribution of Wiener functionals is applied. The effects of…
We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms, focussing on the case of multiple, non-commensurate frequencies. We derive an asymptotic expansion in inverse powers of the…
We obtain the asymptotic expansion of the Voigt functions $K(x,y)$ and $L(x,y)$ for large (real) values of the variables $x$ and $y$, paying particular attention to the exponentially small contributions. A Stokes phenomenon is encountered…
We construct new integral representations for transformations of the ordinary generating function for a sequence, $\langle f_n \rangle$, into the form of a generating function that enumerates the corresponding "square series" generating…