On Multiple Interpolation Functions of the Nِrlund-Type q-Euler Polynomials

and Applied Analysis 3 Remark 1.2. In 1.6 ; we easily see that lim q→ 1 F r q t, x 2 ∞ ∑ m 0 −1 m ( m r − 1 m ) e m x t 2e ∞ ∑ m 0 −1 m ( m r − 1 m ) e 2e 1 et r F r t, x . 1.7 From the above, we obtain generating function of the Nörlund Euler numbers of higher order. That is F r t, x 2e 1 et r ∞ ∑ n 0 E r n x t n! . 1.8 Thus, we have lim q→ 1 E r n,q x E r n x . 1.9 cf. 21 . Hence, we have F r t, x ( 2 et 1 )( 2 et 1 ) · · · ( 2 et 1 ) e 2e ∞ ∑ n1 0 e1 −1 n1 ∞ ∑ n2 0 e2 −1 n2 · · · ∞ ∑ nr 0 er −1 nr 2e ∞ ∑ n1,n2,...,nr 0 −1 n1 n2 ··· nr e n1 n2 ··· nr ∞ ∑ n 0 E r n x t n! . 1.10 We now summarize the results of this paper. In Section 2, we study on modified generating functions of higher-order Nörlund-type q-Euler polynomials and numbers. We obtain some relations related to these numbers and polynomials. In Section 3, we give interpolation functions of the higher order Nörlund-type q-Euler polynomials. In Section 4, we obtain some relations related to he higher order Nörlund-type q-Euler polynomials. In Section 5, we give remarks and observations on an Approximation theory related to Bernoulli and Euler polynomials. 4 Abstract and Applied Analysis 2. Modified Generating Functions of Higher-Order Nörlund-Type q-Euler Polynomials and Numbers In this section we define generating function of modified higher order Nörlund type q-Euler polynomials and numbers, which are denoted by E r n,q x , and E r n,q respectively. We give relations between these numbers and polynomials. We modify 1.6 as follows: F r q t, x F r q ( q−xt, x ) , 2.1 where F r q t, x is defined in 1.6 . From the above we find that F r q t, x ∞ ∑ n 0 q−nxE r n,q x t n! . 2.2 After some elementary calculations, we obtain F r q t, x exp ( x q−xt ) f r q t , 2.3