This paper explores a one-parameter extension of the Hartley kernel expressed as a real combination of two Bessel functions, termed the Hartley–Bessel function. The key feature of the Hartley–Bessel function is derived through a limit transition from the \(-1\) little Jacobi polynomials. The Hartley–Bessel function emerges as an eigenfunction of a first-order difference-differential operator and possesses a Sonin integral-type representation. Our main contribution lies in investigating anovel product formula for this function, which subsequently facilitates the development of innovative generalized translation and convolution structures on the real line. The obtained product formula is expressed as an integral in terms of this function with an explicit non-positive and uniformly bounded measure. Consequently, a non-positivity-preserving convolution structure is established.
