Formula for analytic continuation of the Kampé de Fériet hypergeometric function

We apply the method of Burchnall—Chaundy operators to the study of expansion formulas for the Kampé de Férriet hypergeometric function $F_{1:1;1}^{0:3;3} [x,y]$. Using the obtained operator identities, we derive 14 expansion formulas. A new group of Euler-type integral representations for the Kampé de Férriet hypergeometric function $F_{1:1;1}^{0:3;3} [x,y]$ is found and its analytic continuation is constructed.