On a multivariable extension of the Lagrange–Hermite polynomials

A new family of analytic functions defined by means of Rodrigues type formula

“…If we use the generating relation (18) for Apostol-Bernoulli polynomials by taking a k = 1 k! , µ = 0, ν = 1, we get In a similar manner, choosing s = 1 and Ω µ+νk (y ) = Θ µ+νk (y) in Theorem 8, we obtain the following class of bilinear generating functions for the functions generated by (5).…”

“…In literature, there are numerous investigations to obtain generating functions and recurrence relations satisfied by special functions and polynomials (see [3,4,5,6,7,8,9,10,11,12,13,14,18]). It is possible to derive a recurrence relation by using a generating function.…”

“…, µ = 0, ν = 1 and then taking into account the generating function (5) give the bilinear generating function for the analytic functions Θ n (x) Besides, it is possible to get multivariable function Ω µ+ψk (y 1 , ..., y s ), (s ∈ N) as an appropriate product of several simpler functions for every suitable choice of the coefficients a k (k ∈ N 0 ). Thus, Theorem 8 can be applied in order to derive various families of multilinear and multilateral generating functions for the functions generated by (5).…”

On a Family of Multivariate Modified Humbert Polynomials

“…This case reduces to a multivariable analogue of Lagrange-Hermite polynomials $\begin{array}{c}{\left(\mathrm{1}-x_{\mathrm{1}}{t}_{\mathrm{1}}-x_{\mathrm{2}}{t}_{\mathrm{2}}^{\mathrm{2}}-\cdots -x_r{t}_r^r\right)}^{-\alpha }\\ \hspace{1em}\hspace{1em}=\sum _{n_{\mathrm{normal1}},\dots ,n_r=\mathrm{normal0}}^{\infty }{H}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r},\end{array}$ which is different from that in [6]. …”

Some new results for the multivariable Humbert polynomials

“…Then Lagrange-Hermite and Erkus-Srivastava multivariable polynomials hold the equations, respectively ( [2], [7])…”