A class of completely monotonic functions involving the polygamma functions

Let $\Gamma (x)$ Γ ( x ) denote the classical Euler gamma function. We set $\psi _{n}(x)=(-1)^{n-1}\psi ^{(n)}(x)$ ψ n ( x ) = ( − 1 ) n − 1 ψ ( n ) ( x ) ($n\in \mathbb{N}$ n ∈ N ), where $\psi ^{(n)}(x)$ ψ ( n ) ( x ) denotes the nth derivative of the psi function $\psi (x)=\Gamma '(x)/\Gamma (x)$ ψ ( x ) = Γ ′ ( x ) / Γ ( x ) . For λ, α, $\beta \in \mathbb{R}$ β ∈ R and $m,n\in \mathbb{N}$ m , n ∈ N , we establish necessary and sufficient conditions for the functions $$ L(x;\lambda ,\alpha ,\beta )=\psi _{m+n}(x)-\lambda \psi _{m}(x+ \alpha ) \psi _{n}(x+\beta ) $$ L ( x ; λ , α , β ) = ψ m + n ( x ) − λ ψ m ( x + α ) ψ n ( x + β ) and $-L(x;\lambda ,\alpha ,\beta )$ − L ( x ; λ , α , β ) to be completely monotonic on $(-\min (\alpha ,\beta ,0),\infty )$ ( − min ( α , β , 0 ) , ∞ ) .As a result, we generalize and refine some inequalities involving the polygamma functions and also give some inequalities in terms of the ratio of gamma functions.

Mellin–Barnes integrals and the method of brackets

The method of brackets is a method for the evaluation of definite integrals based on a small number of rules. This is employed here for the evaluation of Mellin–Barnes integral. The fundamental idea is to transform these integral representations into a bracket series to obtain their values. The expansion of the gamma function in such a series constitute the main part of this new application. The power and flexibility of this procedure is illustrated with a variety of examples.

NON-CENTRAL MOMENTS OF THE TRUNCATED NORMAL VARIABLE IN FINANCE

This paper computes the Non-central Moments of the Truncated Normal variable, i.e. a Normal constrained to assume values in the interval with bounds that may be finite or infinite. We define two recursive expressions where one can be expressed in closed form. Another closed form is defined using the Lower Incomplete Gamma Function. Moreover, an upper bound for the absolute value of the Non-central Moments is determined. The numerical results of the expressions are compared and the different behavior for high value of the order of the moments is shown. The limitations to the use of Truncated Normal distributions with a lower negative limit regarding financial products are considered. Limitations in the application of Truncated Normal distributions also arise when considering a CRRA utility function.

Some q-supercongruences related to Swisher’s (H.3) conjecture

We first give a [Formula: see text]-analogue of a supercongruence of Sun, which is a generalization of Van Hamme’s (H.2) supercongruence for any prime [Formula: see text]. We also give a further generalization of this [Formula: see text]-supercongruence, which may also be considered as a generalization of a [Formula: see text]-supercongruence recently conjectured by the second author and Zudilin. Then, by combining these two [Formula: see text]-supercongruences, we obtain [Formula: see text]-analogues of the following two results: for any integer [Formula: see text] and prime [Formula: see text] with [Formula: see text] [Formula: see text] [Formula: see text] which are generalizations of Swisher’s (H.3) conjecture modulo [Formula: see text] for [Formula: see text]. The key ingredients in our proof are the ‘creative microscoping’ method, the [Formula: see text]-Dixon sum, Watson’s terminating [Formula: see text] transformation, and properties of the [Formula: see text]-adic Gamma function.

EXCEL-orientated procedure for calculating the values of special functions with interval argument assigned on the hyperbolic form

Aim of the work is to propose the main terms of the EXCEL-orientated procedures for calculating the values of elementary and special functions with interval argument that is assigned on the hyperbolic form. The results of the work. The methods of presenting the interval values in the hyperbolic form and the rules of addition, subtraction, multiplication, and division of this values were considered. The procedures of calculating the function values, whose arguments can be degenerate or interval values were described. Namely, the direct and the reverse functions of the linear trigonometry, the direct and the reverse functions of the hyperbolic trigonometry, exponential function, arbitrary exponential function and power function, Gamma-function, incomplete Gamma-function, digamma-function, trigamma-function, tetragamma-function, pentagamma-function, Beta-function and its partial derivatives, integral exponential function, integral logarithm, dilogarithm, Frenel integrals, sine integral, cosine integral, hyperbolic sine integral, hyperbolic cosine integral. The basic terms of the EXCEL-orientated procedures for calculating the values of elementary and special functions with interval argument that is assigned on the hyperbolic form were proposed. The numerical examples were provided, that illustrate the application of the proposed methods.

A Note on the Summation of the Incomplete Gamma Function

We examine the improved infinite sum of the incomplete gamma function for large values of the parameters involved. We also evaluate the infinite sum and equivalent Hurwitz-Lerch zeta function at special values and produce a table of results for easy reading. Almost all Hurwitz-Lerch zeta functions have an asymmetrical zero distribution.

The $$\gamma$$ function in quantum theory II. Mathematical challenges and paradoxa

While the square root of Dirac’s $$\delta$$ δ is not defined in any standard mathematical formalism, postulating its existence with some further assumptions defines a generalized function called $$\gamma (x)$$ γ ( x ) which permits a quasi-classical treatment of simple systems like the H atom or the 1D harmonic oscillator for which accurate quantum mechanical energies were previously reported. The so-defined $$\gamma (x)$$ γ ( x ) is neither a traditional function nor a distribution, and it remains to be seen that any consistent mathematical approaches can be set up to deal with it rigorously. A straightforward use of $$\gamma (x)$$ γ ( x ) generates several paradoxical situations which are collected here. The help of the scientific community is sought to resolve these paradoxa.

Series expansion of the Gamma function and its reciprocal

In this paper we give representations for the coefficients of the Maclaurin series for \Gamma(z+1) and its reciprocal (where \Gamma is Euler’s Gamma function) with the help of a differential operator \mathfrak{D}, the exponential function and a linear functional ^{*} (in Theorem 3.1). As a result we obtain the following representations for \Gamma (in Theorem 3.2): \begin{align*} \Gamma(z+1) & = \big(e^{-u(x)}e^{-z\mathfrak{D}}[e^{u(x)}]\big)^{*}, \\ \big(\Gamma(z+1)\big)^{-1} & = \big(e^{u(x)}e^{-z\mathfrak{D}}[e^{-u(x)}]\big)^{*}. \end{align*} Theorem 3.1 and Theorem 3.2 are our main results. With the help of the first theorem we give our approach for finding the coefficients of Maclaurin series for \Gamma(z+1) and its reciprocal in an explicit form.

Quasi-Hopf twist and elliptic Nekrasov factor

We investigate the quasi-Hopf twist of the quantum toroidal algebra of $$ {\mathfrak{gl}}_1 $$ gl 1 as an elliptic deformation. Under the quasi-Hopf twist the underlying algebra remains the same, but the coproduct is deformed, where the twist parameter p is identified as the elliptic modulus. Computing the quasi-Hopf twist of the R matrix, we uncover the relation to the elliptic lift of the Nekrasov factor for instanton counting of the quiver gauge theories on ℝ4× T2. The same R matrix also appears in the commutation relation of the intertwiners, which implies an elliptic quantum KZ equation for the trace of intertwiners. We also show that it allows a solution which is factorized into the elliptic Nekrasov factors and the triple elliptic gamma function.