Derivation Operator And Q-Harmonic Number Identities

On the basis of existing proof techniques of combinatorial identities, several new perfect combinatorial identities are proposed and proved by way of partial fraction and high order derivation. Main points are summarized as follows.1. A q-binominal identity is obtained using q-Chu-Vandermonde convolution, and by taking the higher derivative of the identity with the help of binominal inversion formula, two interesting promotions of Prodinger formula is acquired. Special values of parameters in promotion formulas can generate more interesting identities.2. Inspired by the technique in Chu’s proof of Weideman harmonic identity, two perfect Harmonic identities related to Bell polynomial, 1 1!()ii n n x?? ?? and 1 1!()ii n n x x??? ??? are deduced as promotions of Weideman identity. The values of the parameters in them can be determined by higher derivative and mathematical induction techniques after expanding these two identities by partial fraction method. Special values of parameters in formula ??x x n i ni???1?1)(! will generate many interesting identities.3. Considering the importance of q-harmonic number, two q-Harmonic identities related with Bell polynomial are acquired by q-analogue, which are ?? ??1 1);();(i n n i i qx qq and ?? ?????1 1)1();();(i n nx qx qq i i. The values of the parameters in them can be determined by higher derivative and mathematical induction techniques after expanding these two identities by partial fraction method. Special values of parameters in formula ?? ?????1 1)1();();(i n nx qx qq i i will generate many interesting identities.