1.1 Theoretical background

If H(x, y, t) can be expanded in powers of t in the form (1) where h_n is independent of x and y, and f_n(x) and g_n(x) are different functions, we utilise the language employed by Rainville see p.170 in [1] and call H(x, y, t) a bilateral generating function. A double Chebyshev series is one that has two univariate Chebyshev polynomials T_n,p(α, β) = T_n(α)T_p(β) as products in each term of the series, where T_n(α) = cos(nθ) and α = cos(θ). A contour integral method used by Brafman [2] was used to obtain generating functions satisfying a Rodigues-type formula reducible to the form (2) where a and b are constants, not both zero and F(x) is independent of n and differentiable an arbitrary number of times.

1.2 Recent developments

Mathematicians around the world have remained interested in generating functions from 1965 to present. Publications from India, Japan, Canada, the United States, England, Russia, Germany, and other European nations provide proof of this interest. Three excellent works on special functions were published in 1968 alone. Each of these took into account the idea of the generating function from a group-theoretic perspective. They were authored by American Willard Miller, Jr. [3], Canadian James D. Talman [4], and Russian N.J. Vilenkin [5]. (V. N. Singh translated Vileqkin’s novel into English). Publications on generating functions by N. A. AI-Salam and W. A. AI-Salam [6], J. W. Brown [7], and L. Carlitz [8] are only a few of the many intriguing ones along with more recent interesting publications. The Special Functions Section of the Mathematical Reviews contains a wealth of references to publications on generating functions. Literature on Chebyshev polynomials are presented in great detail in section 18 in [9]. These polynomials have well known generating functions in current literature and are used widely in all areas of mathematics and science. A particular popular application of these generating functions is in the solution of partial differential equations. Famous mathematicians like Lanczos [10] used the bivariate form of these generating functions along with their strong convergence properties in solving ordinary differential equations. In the work by Mason [11] the bivariate form of Chebyshev polynomials was employed in studying polynomial approximation. Examples of bilateral generating functions and their derivations are detailed in chapter 1 in McBride [12] and in the work by Mohammad [13]. The Chebyshev polynomial has also been studied and used in numerical solutions of initial boundary equations [14–16]. Generating functions involving special functions have been studied in the work by Meena et al. [17–20]. Solutions of partial differential equations in terms of two-dimensional Cbebyshev series have been studied with possible applications to the eigenvalue problem for a vibrating L-shaped membrane. In this work we provide a closed form solution to the bilateral generating function featuring the double summation of Chebyshev polynomials expressed in terms of the incomplete gamma function similar to the forms given in the works [12, 13]. In section 3, the Chebyshev contour integral formula is derived. In section 4, we give a detailed account of the incomplete gamma including integral and summation definitions. In section 5, the contour integral representations for the incomplete gamma function are derived. section 6, we formulate the main theorem and deduce a few propositions and examples in terms of constant and composite functions. In section 7, we look at the limiting case of the difference of the bivariate Chebyshev polynomials expressed in terms of the incomplete gamma function. Our preliminaries start with the contour integral method [21], applied to the formula for the Chebyshev generating function given by equation (18.12.8) in [9]. Let a, α, β, k and w be general complex numbers, n ∈ [0, ∞) and p ∈ [0, ∞), where the contour integral form of the Chebyshev generating function is given by (3) where |Re(w)| < 1. We will use Eq (3) to derive equivalent sums for the left-hand side and a Special function form for the right-hand side. The derivation of the summation follows the method used by us in [21] which involves Cauchy’s integral formula. The generalized Cauchy’s integral formula is given by (4) where C is in general, an open contour in the complex plane where the bilinear concomitant has the same value at the end points of the contour. This method involves using a form of Eq (4) then multiply both sides by a function, then take a definite sum of both sides. This yields a definite sum in terms of a contour integral. A second contour integral is derived by multiplying Eq (4) by a function and performing some substitutions so that the contour integrals are the same.

4.1 Derivation of the right-hand side contour integral representations

In this sub-section we derive the Incomplete Gamma function in terms of the contour integral representation for the right-hand side of Eq (3). These formulae are achieved by analyzing the right-hand side of Eq (3) and decomposing the quotient of polynomials into partial fractions and applying a definite integral.

7.1 Initial equations

Here we identify the equation we would like to find the new representation for given by; (39)

Next we increase the factorial by one and assign a general function g(z) to be solved given by; (40)

Next we take the difference of Eqs (39) and (40) to get; (41)

Next we take the derivative of Eq (40) such that the left-hand side is equal to the left-hand side of Eq (41) to get; (42)

Since the left-hand sides of Eqs (41) and (42) are the same we may equate the right-hand sides and solve the ordinary differential equation given by; (43)

Next we solve the differential equation for the function g(z), with independent variable z given by; (44)

Next we apply the initial condition of g(0) = 0 and simplify to get; (45)

7.2 Examples of generating functions

Repeating the method above the following Chebyshev generating functions are derived; (46) (47) (48) (49)