Reformulated q-middle convolution yields additive composition properties

What the study found: The authors reformulate q-convolution and q-middle convolution, and they introduce q-analogues of addition related to gauge transformation. A key merit of the reformulation is that composition of two q-middle convolutions is additive.
Why the authors say this matters: The authors suggest that the reformulation provides a useful structure for working with q-middle convolution and related q-difference equations. They also present it as a way to obtain solutions to certain third-order linear q-difference equations.
What the researchers tested: The paper studies reformulations of q-convolution and q-middle convolution introduced by Sakai and Yamaguchi. It also develops q-analogues of addition and examines Jackson integrals associated with q-convolution.
What worked and what didn't: The authors obtain sufficient conditions for the Jackson integrals to converge and to satisfy the q-difference equation associated with q-convolution. They also present several third-order linear q-difference equations and give solutions using q-middle convolution and the q-analogues of addition.
What to keep in mind: The abstract does not describe any limitations, negative results, or comparisons with other methods. The summary is limited to the reformulation, convergence conditions, and example equations described in the abstract.

Key points

• The paper reformulates q-convolution and q-middle convolution.
• It introduces q-analogues of addition related to gauge transformation.
• A stated merit is that composition of two q-middle convolutions is additive.
• The authors give sufficient conditions for certain Jackson integrals to converge and satisfy the associated q-difference equation.
• Several third-order linear q-difference equations are presented with solutions obtained by the methods in the paper.