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6 EXERCISES 305 89 Develop a general theory of the solutions to the two-parameter recurrence = (an+ @+y) +(a’n+/3’k+y’)...

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6 EXERCISES 305

89 Develop a general theory of the solutions to the two-parameter recurrence = (an+ @+y) +(a’n+/3’k+y’)

+[n=k=OI,

forn,k30,

assuming that [:I = 0 w h en n < 0 or k < 0. (Binomial coefficients, Stirling numbers, Eulerian numbers, and the sequences of exercises 17 and 31 are special cases.) What special values (LX, fl,r, CX’, fi’,~‘) yield “fundamental solutions” in terms of which the general solution can be expressed?