The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions
The authors consider a new expansion of an analytic function \(f(z)\) in \(\mathbb R\) in terms of a power series \(\theta(t) = tq(t)\), where \(q(t)\) is any regular function and is equal to the quadratic function \(\left[ {\left( {z - z_1 } \right)\left( {z - z_2 } \right)} \right], z_1 \neq z_2.\) It is shown that the coefficient of \((z - z_1 )^n (z - z_2 )^n\) in the power series of the function \((z - z_1 )^\alpha (z - z_2 )^\beta f(z)\) is \[ D_{z_1 - z_2 }^{ - \alpha + n} \left[ {f(z_1 )(z_1 - z_2 )^{\beta - n - 1} (z_1 - z_2 + z - w)} \right]\left. {} \right|_{w = z_1 } /\Gamma (1 - \alpha + n), \] where \(\alpha\) and \(\beta\) are arbitrary complex numbers. The derivation of the main results makes use of the multi-loop contour integration (Pochhammer contour), and several special forms are mentioned giving some new identities involving special functions and integrals. The paper contains a comprehensive list of references which would be of immense benefit to the interested reader.
scientific article; zbMATH DE number 5163959
scientific article