Pieri-type multiplication formula for quantum Grothendieck polynomials

The purpose of this paper is to prove a Pieri-type multiplication formula for quantum Grothendieck polynomials, which was conjectured by Lenart-Maeno. This formula would enable us to compute explicitly the quantum product of two arbitrary (opposite) Schubert classes in the (small) quantum $K$-theory ring $QK(F{l}_n)$ of the (full) flag manifold $F{l}_n$ of type $A_{n-1}$ on the basis of the fact that quantum Grothendieck polynomials represent (opposite) Schubert classes in $QK(F{l}_n)$.