Abstract
In the article, in the positive domain \(\Omega=\big{\{}(x,y,t):\,x>0,\,y>0,\,t>0\big{\}}\) we consider a degenerate third-order differential equation of the form \(x^{n}y^{m}\,u_{t}=t^{k}y^{m}\,u_{xxx}+t^{k}x^{n}\,u_{yyy}\) , \(m,n,k={\textrm{const}}>0\) . Nine partial solutions of this equation are expressed through the Campe de Feriet hypergeometric functions \(F_{0;2;2}^{1;0,0}[x,y]\) . By generalizing the operator method of J.L. Burchnall and T.W. Chaundy, one-dimensional reciprocal symbolic operators are introduced. Using Burchnall–Chaundy operators, decomposition formulas and integral representations for the Campe de Feriet hypergeometric function \(F_{0;2;2}^{1;0,0}[x,y]\) are obtained.
