## Properties of Ultra Gamma Function by Kuldeep Singh Gehlot

In this paper we study the integral of type
\begin{equation}
{}_{\delta , a}\Gamma_{\rho , b}(x) = \Gamma(\delta , a; \rho , b)(x) = \int\limits_{0}^{\infty} t^{x-1} \mathrm{exp}\left( -\frac{t^{\delta}}{a}-\frac{1}{b t^{\rho}} \right) dt
\end{equation}
Different authors called this integral by different names like ultra gamma function, generalized gamma function, Kratzel integral, inverse Gaussian integral, reaction-rate probability integral, Bessel integral etc. We prove several identities and recurrence relation of above said integral, we called this integral as Four Parameter Gamma Function. Also we evaluate relation between Four Parameter Gamma Function, p-k Gamma Function and Classical Gamma Function. With some conditions we can evaluate Four Parameter Gamma Function in term of Hypergeometric function.

The entire paper is available here.

## Legendre-type relations for generalized complete elliptic integrals by Shingo Takeuchi

Legendre’s relation for the complete elliptic integrals of the first and second kinds is generalized. The proof depends on an application of the generalized trigonometric functions and is alternative to the proof for Elliott’s identity.

The entire paper is available here.

## On The Extended Incomplete Pochhammer Symbols and Hypergeometric Functions by Rakesh Kumar Parmar, R.K. Raina

In this paper, we first introduce certain forms of extended incomplete Pochhammer symbols which are then used to define families of extended incomplete generalized hypergeometric functions. For these functions, we investigate various properties including the integral representations, derivative formula, certain generating function and fractional integrals (and derivatives) relationships. Some special cases of the main results are also deduced.

The entire paper is available here.

## A Collection of Free Math Books from Folkscanomy Mathematics: Books of a Mathematic Nature

See Folkscanomy Mathematics: Books of a Mathematic Nature. Here, you can find such books as The Theory of Functions by E.C. Titchmarsh and A Treatise on the Theory of Bessel Functions by G.N. Watson.

## Some formulae for products of Fubini polynomials with applications by Levent Kargın

In this paper we evaluate sums and integrals of products of Fubini polynomials and have new explicit formulas for Fubini polynomials and numbers. As a consequence of these results new explicit formulas for p-Bernoulli numbers and Apostol-Bernoulli functions are given. Besides, integrals of products of Apostol-Bernoulli functions are derived.

The entire paper is available here.

## Higher order generalized geometric polynomials by Levent Kargin, Bayram Çekim

According to generalized Mellin derivative (Kargin), we introduce a new family of polynomials called higher order generalized geometric polynomials. We obtain some properties of them.We discuss their connections to degenerate Bernoulli and Euler polynomials. Furthermore, we find new formulas for the Carlitz’s (Carlitz) and Howard’s (Howard2) finite sums. Finally, we evaluate several series in closed forms, one of which has the coefficients include values of the Riemann zeta function. Moreover, we calculate some integrals in terms of generalized geometric polynomials.

The entire paper is available here.

## Certain unified integration formulas associated with generalized k-Bessel function by G. Rahman, K. S. Nisar, S. Mubeen, M. Arshad

Our purpose in this present paper is to investigate generalized integration formulas containing the generalized k-Bessel function $$W^{k}_{v,c}(z)$$ to obtain the results in representation of Wright-type function. Also, we establish certain special cases of our main result.

The entire paper is available here.

## Some integrals involving generalized k-Struve functions by K.S. Nisar, S.R. Mondal

The close form of some integrals involving recently developed generalized k-Struve functions is obtained. The outcome of these integrations is expressed in terms of generalized Wright functions. Several special cases are deduced which lead to some known results.

The entire paper is available here.

## Extension of a factorization method of nonlinear second order ODEs with variable coefficients by H.C. Rosu, O. Cornejo-Perez, M. Perez-Maldonado, J.A. Belinchon

The factorization of nonlinear second-order differential equations proposed by Rosu and Cornejo-Perez in 2005 is extended to equations containing quadratic and cubic forms in the first derivative. A few illustrative physics examples are provided.

The entire paper is available here.

## Integration by differentiation: new proofs, methods and examples by Ding Jia, Eugene Tang, Achim Kempf

Recently, new methods were introduced which allow one to solve ordinary integrals by performing only derivatives. These studies were originally motivated by the difficulties of the quantum field theoretic path integral, and correspondingly, the results were derived by heuristic methods. Here, we give rigorous proofs for the methods to hold on fully specified function spaces. We then illustrate the efficacy of the new methods by applying them to the study of the surprising behavior of so-called Borwein integrals.

The entire paper is available here.