Concrete mathematics : a foundation for computer science

236 BINOMIAL COEFFICIENTS
57 Use Gosper’s method to find a constant 8 such that
is summable in hypergeometric terms.
58 If m and n are integers with 0 6 m 6 n, let
Tm,n =
Find a relation between T,,,n and T,-1 ,+I, then solve your recurrence
by applying a summation factor.
Exam problems
59
60
61
62
63
Find a closed form for
when m and n are positive integers.
Use Stirling’s approximation (4.23) to estimate (“,‘“) when m and n are
both large. What does your formula reduce to when m = n?
Prove that when p is prime, we have
for all nonnegative integers m and n.
Assuming that p is prime and that m and n are positive integers, determine
the value of (,‘$‘) mod p2. Hint: You may wish to use the following
generalization of Vandermonde’s convolution:
k+k&+k JI:)(~)-~(~) = (r’+r2+i-~+Tm)*
1 2 m
Find a closed form for
given an integer n >, 0.