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

n this paper, we present $q$-Bernoulli and $q$-Euler polynomials generated by the third Jackson $q$-Bessel function to construct new types of $q$-Lidstone expansion theorem. We prove that the entire function may be expanded in terms of $q$-Lidstone polynomials which are $q$-Bernoulli polynomials and the coefficients are the even powers of the $q$-derivative $\frac{\delta_q f(z)}{\delta_q z}$at$0$and$1$\mathrm{.}Theotherformsexpandthefunctionin$q$-Lidstonepolynomialsbasedon$q$-Euler polynomials and the coefficients contain the even and odd powers of the $q$-derivative $\frac{\delta_q f(z)}{\delta_q z}$.