  An L-basis associated to a linear second-order ordinary differential operator
L is an infinite sequence of functions {\phi_k}_{k=0}^{\infty} such that
L\phi_k=0 for k=0,1, L\phi_k=k(k-1)\phi_{k-2}, for k=2,3,... and all \phi_k
satisfy certain prescribed initial conditions. We study the transmutation
operators related to L in terms of the transformation of powers of the
independent variable {(x-x_{0})^k}_{k=0}^{\infty} to the elements of the
L-basis and establish a precise form of the transmutation operator realizing
this transformation. We use this transmutation operator to establish a
completeness of an infinite system of solutions of the stationary Schr\"odinger
equation from a certain class. The system of solutions is obtained as an
application of the theory of bicomplex pseudoanalytic functions and its
completeness was a long sought result. Its use for constructing reproducing
kernels and solving boundary and eigenvalue problems has been considered even
without the required completeness justification. The obtained result on the
completeness opens the way for further development and application of the tools
of pseudoanalytic function theory.
