The q-Lidstone series involving q-Bernoulli and q-Euler polynomials generated by the third Jackson q-Bessel function

Abstract

We present q-Bernoulli and q-Euler polynomials generated by the third Jackson q-Bessel function. We derive new q-analogues of the Lidstone expansion theorem. First, we expand a certain class of entire functions in terms of the q-Bernoulli polynomials we introduce. The coefficients of the polynomials are the even powers of the symmetric q-derivative $\frac{{\mathit{\delta }}_{q}f(z)}{{\mathit{\delta }}_{q}z}$ at 0 and 1. The other forms expand a certain class of functions in the q-Euler polynomials, where the coefficients of the expansions are the even and odd powers of the symmetric q-derivative $\frac{{\mathit{\delta }}_{q}f(z)}{{\mathit{\delta }}_{q}z}$ at 0 and 1.