On a q-analogue for Bernoulli numbers

On a class of generalized q-Bernoulli and q-Euler polynomials

“…The formulas ( 16)-( 18) are q-extension of the Cheon's main result [5]. Notice that B 1,q = − 1 [2] q , see [26],…”

“…respectively. Note that the q-Bernoulli numbers B n,q are defined and studied in [26]. The aim of the present paper is to obtain some results for the above defined q-Bernoulli and q-Euler polynomials.…”

Approximation Properties of q-Bernoulli Polynomials

“…when q → 1 from the right side, we have an ordinary form of this relation. For the another forms of sum of power, see [22] Example 8 Let f (x) = e −x q = 1 E x q then this function decreases so rapidly with x such that all normal derivatives approach zero as x → ∞. For comfirming this, let us mention that E x q , for some fixed…”

“…When → 1 from the right side, we have an ordinary form of this relation. For the another forms of sum of power, see [15]. …”

“…Example 7 Let f (x) = x s where s is a positive integer, then we can apply this function at (22) where a = 0…”