Problem 1) Prove the identity
What do we get if ?
Hint: Count the number of mappings of a set of k given elements onto a set of n given elements.
Problem 2) Prove the following Abel identities:
Hint: To prove problem 2a, differentiate both sides with respect to y, and use induction on n. The other two sub-problems will follow from this sub-problem a.
Problem 3) Let
Prove that .
Find other examples of such sequences of polynomials.
Problem 4) Evaluate the determinants:
where denotes the greatest common divisor of i and j.
Problem 5) The number of non-congruent triangles with circumference , and integer sides is equal to the number of non-congruent triangles with circumference and integer sides. This number is also equal to the number of partitions of n into exactly three terms. Determine the number.
More conundrums to follow…(no hints for problems 4 and 5 above ! :-))