Entire Solutions of Painlevé V-Type Differential- Difference Equation

This research focuses on the following equation: (g¯g−1)(gg̲−1)+B(z)g′(z)g(z)=R(z,g)=P(z,g)Q(z,g), where P(z,g) and Q(z,g) represent polynomials in g, possessing rational coefficients in z and lacking shared roots, B(z) is rational, and g¯≡g(z+1),g̲≡g(z−1). Reduced forms arise when this equation permits an entire solution with a hyper-order strictly below one. This paper investigates the forms and value distribution of entire solutions to the reduced Painlevé-type differential-difference equation. Furthermore, the existence of entire solutions to the above equation is examined when degg(P)≥5 and degg(Q)=0 or degg(Q)≥1 and Q(z,0)≢0.