I am currently proving that $\pi$ is irrational, and I am basing from Cartwright's proof. I let \begin{equation*} A_k(x) = \int_{-1}^{1} (1-y^2)^k \cos(xy)dy \end{equation*} where $k \in \{0\} \cup \mathbb{Z}^+$. I then used integration by parts and got \begin{equation*} x^2A_k(x) = 2k(2k-1)A_{k-1}(x) - 4k(k-1)A_{k-2}(x), k \geq 2. \end{equation*} Next, I let $B_k(x) = x^{2k+1}A_k(x)$ for all nonnegative integer $k$.
I have already shown \begin{align*} B_k(x) &= 2k(2k-1)B_{k-1}(x) - 4k(k-1)x^2B_{k-2}(x)\\ B_0(x) &= 2\sin x\\ B_1(x) &= 4\sin x -4x\cos x\\ \end{align*} I am struggling to show \begin{equation} B_k(x) = k!(M_k(x)\sin x + N_k(x)\cos x) \end{equation} where $M_k(x)$ and $N_k(x)$ are polynomials in $x$ with integer coefficients and of degree less than or equal to $k$ for all nonnegative integer $k$. How do I show this? Based on what I have seen online, I have to use mathematical induction and show it holds for $k+1$ i.e., \begin{equation*} B_{k+1}(x) = (k+1)!(M_{k+1}(x)\sin x + N_{k+1}(x)\cos x) \end{equation*}
Notation: $\Bbb Z[x]$ is a standard notation for the set of polynomials in the free variable $x,$ with integer co-efficients.
Let $S(k)$ be the statement $T(k)\land T(k+1)$ where $T(k)$ is the statement that $B_k(x)=k![M_k\sin x +N_k\cos x]$ where $M_k, N_k\in \Bbb Z[x]$ with $\max (\deg M_k, \deg N_k)\le k.$
You have $S(0).$ Now prove that $\forall k\in \Bbb N_0\,[S(k)\implies S(k+1)].$
Note that $S(k)\implies T(k+1)$ while $[S(k+1)]\iff [T(k+1)\land T(k+2)] .$ So to prove $S(k)\implies S(k+1),$ it suffices to prove that $S(k)\implies T(k+2).$