Abstract
In this paper, we investigate the structural and analytic properties of the two-parameter telephone polynomials. Starting from the exponential generating function and the recurrence relation, we derive an explicit formula, an operational representation, and the raising and lowering operators of the family. We then examine its differential identities, a Rodrigues-type formula, several integral formulas, and its connections with some special polynomial families. For the Gaussian weight we obtain a closed form for the weighted inner product, together with a contour orthogonality relation on the imaginary axis. Finally, we investigate their zeros for some selected parameter values and present graphical illustrations of their distribution.