Find the generating functions of the sequences of Chebyshev polynomials of the first and second kind: \[F_T(x,z) = \sum_{n=0}^{\infty}T_n(x)z^n;\quad F_U(x,z) = \sum_{n=0}^{\infty}U_n(X)z^n.\]
Definitions of Chebyshev polynomials can be found in the handbook.
We denote by \(P_{k, l}(n)\) the number of partitions of the number \(n\) into at most \(k\) terms, each of which does not exceed \(l\). Prove the equalities:
a) \(P_{k, l}(n) - P_{k, l-1}(n) = P_{k-1, l}(n-l)\);
b) \(P_{k, l}(n) - P_{k-1, l} (n) = P_{k, l-1}(n-k)\);
c) \(P_{k, l}(n) = P_{l, k} (n)\);
d) \(P_{k, l}(n) = P_{k, l} (kl - n)\).
Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.
Find the coefficient of \(x\) for the polynomial \((x - a) (x - b) (x - c) \dots (x - z)\).
The following words/sounds are given: look, yar, yell, lean, lease. Determine what will happen if the sounds that make up these words are pronounced in reverse order.
Prove that the polynomial \(P (x)\) is divisible by its derivative if and only if \(P (x)\) has the form \(P(x) = a_n(x - x_0)^n\).
Prove that for \(n > 0\) the polynomial \[P (x) = n^2x^{n + 2} - (2n^2 + 2n - 1) x^{n + 1} + (n + 1)^2x^n - x - 1\] is divisible by \((x - 1)^3\).
Prove that for \(n> 0\) the polynomial \(x^{2n + 1} - (2n + 1)x^{n + 1} + (2n + 1)x^n - 1\) is divisible by \((x - 1)^3\).
Prove that the polynomial \(P (x) = a_0 + a_1x + \dots + a_nx^n\) has a number \(-1\) which is a root of multiplicity \(m + 1\) if and only if the following conditions are satisfied: \[\begin{aligned} a_0 - a_1 + a_2 - a_3 + \dots + (-1)^{n}a_n &= 0,\\ - a_1 + 2a_2 - 3a_3 + \dots + (-1)^{n}na_n &= 0,\\ \dots \\ - a_1 + 2^{m}a_2 - 3^{m}a_3 + \dots + (-1)^{n}n^{m}a_n &= 0. \end{aligned}\]
A class contains 33 pupils, who have a combined age of 430 years. Prove that if we picked the 20 oldest pupils they would have a combined age of no less than 260 years. The age of any given pupil is a whole number.