Problems

Liouville’s discrete theorem. Let $f(x,y)$ be a bounded harmonic function (see the definition in problem number 11.28), that is, there exists a positive constant $M$ such that $\mathrm{\forall }(x,y)\in {\mathbb{Z}}^2$ $|f(x,y)|\le M$. Prove that the function $f(x,y)$ is equal to a constant.

Definition. The sequence of numbers $a_0,a_1,\dots ,a_n,\dots$, which, with the given $p$ and $q$, satisfies the relation $a_{n+2}=p{a}_{n+1}+q{a}_n$ ($n=0,1,2,\dots$) is called a linear recurrent sequence of the second order.

The equation $$x^2-px-q=0$$ is called a characteristic equation of the sequence $\{a_n\}$.

Prove that, if the numbers $a_0$, $a_1$ are fixed, then all of the other terms of the sequence $\{a_n\}$ are uniquely determined.

A frog jumps over the vertices of the triangle $ABC$, moving each time to one of the neighbouring vertices.

How many ways can it get from $A$ to $A$ in $n$ jumps?

The frog jumps over the vertices of the hexagon $ABCDEF$, each time moving to one of the neighbouring vertices.

a) How many ways can it get from $A$ to $C$ in $n$ jumps?

b) The same question, but on condition that it cannot jump to $D$?

c) Let the frog’s path begin at the vertex $A$, and at the vertex $D$ there is a mine. Every second it makes another jump. What is the probability that it will still be alive in $n$ seconds?

d)* What is the average life expectancy of such frogs?

Let $(1+\sqrt{2}+\sqrt{3}{)}^n=p_n+q_n\sqrt{2}+r_n\sqrt{3}+s_n\sqrt{6}$ for $n\ge 0$. Find:

a) $\underset{n\to \mathrm{\infty }}{lim}\frac{p_n}{q_n}$; b) $\underset{n\to \mathrm{\infty }}{lim}\frac{p_n}{r_n}$; c) $\underset{n\to \mathrm{\infty }}{lim}\frac{p_n}{s_n}$;

Find the generating functions of the sequences of Chebyshev polynomials of the first and second kind: $$F_T(x,z)=\sum _{n=0}^{\mathrm{\infty }}{T}_n(x)z^n;F_U(x,z)=\sum _{n=0}^{\mathrm{\infty }}{U}_n(X)z^n.$$

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.