Estimates of Derivatives in Sobolev Spaces in Terms of Hypergeometric Functions

The paper deals with sharp estimates of derivatives of intermediate order $k\le n-1$ in the Sobolev space $\mathring W^n_2[0;1]$, $n\in\mathbb N$. The functions $A_{n,k}(x)$ under study are the smallest possible quantities in inequalities of the form
$$|y^{(k)}(x)|\le A_{n,k}(x)\|y^{(n)}\|_{L_2[0;1]}.$$
The properties of the primitives of shifted Legendre polynomials on the interval $[0;1]$ are used to obtain an explicit description of these functions in terms of hypergeometric functions. In the paper, a new relation connecting the derivatives and primitives of Legendre polynomials is also proved.