New exact solutions of the discrete fourth Painlevé equation

In this paper we derive a number of exact solutions of the discrete equation

where 𝑧_𝑛 = 𝑛𝛿 and 𝜂, 𝛿, 𝜇 and 𝛾 are constants. In an appropriate limit (1) reduces to the fourth Painlevé (PIV) equation

where α and β are constants and (1) is commonly referred to as the discretised fourth Painlevé equation. A suitable factorisation of (1) facilitates the identification of a number of solutions which take the form of ratios of two polynomials in the variable 𝑧_𝑛. Limits of these solutions yield rational solutions of PIV (2). It is also known that there exist exact solutions of PIV (2) that are expressible in terms of the complementary error function and in this article we show that a discrete analogue of this function can be obtained by analysis of (1).