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A new product formula involving Bessel functions. (English) Zbl 1515.33008

Considering the generalized Hankel function \[ B_{\lambda}^{\kappa,n}(x) = j_{\kappa n-\frac{n}{2}}\left( n| \lambda x | ^{\frac{1}{n}}\right) + (-i)^n \left(\frac{n}{2}\right)^n \frac{\Gamma\left(\kappa n-\frac{n}{2}+1 \right)}{\Gamma\left( \kappa n +\frac{n}{2}+1 \right)} \lambda x j_{\kappa n + \frac{n}{2}} \left( n| \lambda x | ^{\frac{1}{n}}\right) \] where \(j_{\alpha}\) is the normalized Bessel function of first kind and order \(\alpha >-\frac{1}{2}\), the authors prove an integral expression for the product \(u^{n} j_{\alpha +n}(u) j_{\alpha}(v) \), \(u,v \in ]0,+\infty[\) and \( n \in \mathbb{N}\).
As a consequence, among others, the following interesting formula is established. \[ B_{\lambda}^{\kappa,n}(x) B_{\lambda}^{\kappa,n}(y) = \int_{\mathbb{R}} B_{\lambda}^{\kappa,n}(z) d \nu_{x,y}^{\kappa,n}(z),\quad \lambda , x, y \in \mathbb{R} \] where \[ d \nu_{x,y}^{\kappa , n}(z) = \begin{cases} \mathcal{K}_{\kappa, n }(x,y,z) d \mu_{\kappa , n}(z) &\text{if } xy \neq 0 \\ d\delta_{x}(z) &\text{if } y=0\\ d\delta_{y}(z) &\text{if } x=0 \end{cases} \] and \(d \mu_{\kappa , n }(x) = \left( M_{\kappa , n} \right)^{-1} | x | ^{2\kappa + \frac{2}{n}-2} dx\), with \(M_{\kappa , n} = 2 \left( \frac{2}{n} \right)^{\kappa n - \frac{n}{2}} \Gamma\left( \kappa n +1 - \frac{n}{2} \right)\), and considering a certain \(\mathcal{K}_{\kappa, n }(x,y,z)\) defined by the Gegenbauer polynomials and \(M_{\kappa , n}\).

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C05 Classical hypergeometric functions, \({}_2F_1\)
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42A85 Convolution, factorization for one variable harmonic analysis
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References:

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