Academic literature on the topic 'A Pochhammer symbol in Applied Mathematics'

<strong>Abstract: </strong>In many areas of applied mathematics, various types of Special functions have become essential tools for Scientists and engineers. Both Beta and Gamma functions are very important in calculus as complex integrals can be moderated into simpler form. In physics and engineering problems require a detailed knowledge of applied mathematics and an understanding of special functions such as gamma and beta functions. The topic of special functions is very important and it is constantly expanding with the existence of new problems in the applied Sciences in this article, we describe the basic theory of gamma and beta functions, their connections with each other and their applicability to engineering problems.to compute and depict scattering amplitude in Reggae trajectories. Our aim is to illustrate the extension of the classical beta function has many uses. It helps in providing new extensions of the beta distribution, providing new extensions of the Gauss hyper geometric functions and confluent hyper geometric function and generating relations, and extension of Riemann-Lowville derivatives. In this Article, we develop some elementary properties of the beta and gamma functions. We give more than one proof for some results. Often, one proof generalizes and others do not. We briefly discuss the finite field analogy of the gamma and beta functions. These are called Gauss and Jacobi sums and are important in number theory. We show how they can be used to prove Fermat's theorem that a prime of the form 4n + 1 is expressible as a sum of two squares. We also treat a simple multidimensional extension of a beta integral.

Given two polynomial sets $\{ P_n(x) \}_{n\geq 0},$ and $\{ Q_n(x) \}_{n\geq 0}$ such that $\deg ( P_n(x) ) = \deg ( Q_n(x) )=n.$ The so-called connection problem between them asks to find coefficients $\alpha_{n,k}$ in the expression $\displaystyle Q_n(x) =\sum_{k=0}^{n} \alpha_{n,k} P_k(x).$ The connection problem for different types of polynomials has a long history, and it is still of interest. The connection coefficients play an important role in many problems in pure and applied mathematics, especially in combinatorics, mathematical physics and quantum chemical applications. For the particular case $Q_n(x)=x^n$ the connection problem is called the inversion problem associated to $\{P_n(x)\}_{n\geq 0}.$ The particular case $Q_n(x)=P'_{n+1}(x)$ is called the derivative connecting problem for polynomial family $\{ P_n(x) \}_{n\geq 0}.$ In this paper, we give a closed-form expression of the inversion and the derivative coefficients for hypergeometric polynomials of the form $${}_2 F_1 \left[ \left. \begin{array}{c} -n, a \\ b \end{array} \right | z \right], {}_2 F_1 \left[ \left. \begin{array}{c} -n, n+a \\ b \end{array} \right | z \right], {}_2 F_1 \left[ \left. \begin{array}{c} -n, a \\ \pm n +b \end{array} \right | z \right],$$ where $\displaystyle {}_2 F_1 \left[ \left. \begin{array}{c} a, b \\ c \end{array} \right | z \right] =\sum_{k=0}^{\infty} \frac{(a)_k (b)_k}{(c)_k} \frac{z^k}{k!},$ is the Gauss hypergeometric function and $(x)_n$ denotes the Pochhammer symbol defined by $$\displaystyle (x)_n=\begin{cases}1, n=0, \\x(x+1)(x+2)\cdots (x+n-1) , n&gt;0.\end{cases}$$ All polynomials are considered over the field of real numbers.

Abstract In this paper, we obtain analytical solutions of an unsolved integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} of Srinivasa Ramanujan [S. Ramanujan, Some definite integrals connected with Gauss’s sums, Mess. Math. 44 1915, 75–86] with suitable convergence conditions in terms of Meijer’s G-function of one variable, by using Mellin–Barnes type contour integral representations of the sine function, Laplace transform method and some algebraic properties of Pochhammer’s symbol. Also, we have given some generalizations of Ramanujan’s integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} in the form of integrals ℧ S * ⁢ ( υ , b , c , λ , y ) {\mho_{S}^{*}(\upsilon,b,c,\lambda,y)} , Ξ S ⁢ ( υ , b , c , λ , y ) {\Xi_{S}(\upsilon,b,c,\lambda,y)} , ∇ S ⁡ ( υ , b , c , λ , y ) {\nabla_{S}(\upsilon,b,c,\lambda,y)} and ℧ S ⁢ ( υ , b , λ , y ) {\mho_{S}(\upsilon,b,\lambda,y)} with suitable convergence conditions and solved them in terms of Meijer’s G-functions. Moreover, as applications of Ramanujan’s integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} , the three new infinite summation formulas associated with Meijer’s G-function are obtained.