Differential recurrence formulae for orthogonal polynomials

Part I - By combining a general 2nd-order linear homogeneous ordinary differential equation with the three-term recurrence relation possessed by all orthogonal polynomials, it is shown that sequences of orthogonal polynomials which satisfy a differential equation of the above mentioned type necessarily have a differentiation formula of the type: gn(x)Y'n(x)=fn(x)Yn(x)+Yn-1(x). Part II - A recurrence formula of the form: rn(x)Y'n(x)+sn(x)Y'n+1(x)+tn(x)Y'n-1(x)=0, is derived using the result of Part I.