Applications of the operator H(α,β)H(α,β) to the Humbert double hypergeometric functions

By making use of some techniques based upon certain new inverse pairs of symbolic operators, the authors investigate several decomposition formulas associated with Humbert hypergeometric functions Φ1Φ1, Φ2Φ2, Φ3Φ3, Ψ1Ψ1, Ψ2Ψ2, Ξ1Ξ1 and Ξ2Ξ2. These operational representations are constructed and applied in order to derive the corresponding decomposition formulas. With the help of these inverse pairs of symbolic operators, as many as 34 decomposition formulas are found. Euler type integrals connected with Humbert functions are also presented.