In this work, polynomials and the finite q-exponential operator are constructed. The is used to combine an proof of generating function with its extension, Mehler's formula extension Roger's for . Rogers Al-Salam-Carlitz deduced by giving special values

The complexity of a lattice polynomial is defined inductively, with variables having complexity 0. If p=plv'"vOk or p=plA'''/XOk is the canonical expression of the polynomial O, then the complexity c(p) = l+max{c(p~):l<-i-k}. An n-k lattice inclusion is an inclusion of the form p <-o-with c(p)<-n and c(o-)-k. In this note we use the main result of Day [1] to show that if all the congruence latt...

In this paper we determine an upper bound for the covering radius of a q-ary MacDonald codeCk;u(q). Values of nq(4; d), the minimal length of a 4-dimensional q-ary code with minimum distance d is obtained for d = q 1 and q 2. These are used to determine the covering radius of C3;1(q),C3;2(q) and C4;2(q).

As a continuation of Maruta [Finite Fields Appl. 10 (2004), 674–685], we investigate the extendability of [n, k, d]q codes with d ≡ −2 (mod q) whose weights are congruent to 0, −1 or −2 (mod q) for even q ≥ 4. We show that such codes are extendable for all even q ≥ 8, giving a new extension theorem for [n, k, d]q codes with gcd(d, q) = 2. We also consider the extendability of such codes for q = 4.

For d≥3, we prove that time-inhomogeneous stochastic differential equations driven by additive noises with drifts in critical Lebesgue space Lq([0,T];Lp(Rd)), where (p,q)∈(d,∞]×[2,∞) and d∕p+2∕q=1, or (p,q)=(d,∞) divb∈L∞([0,T];Ld∕2+ε(Rd)), are well-posed. The weak uniqueness is obtained solving corresponding Kolmogorov backward some second-order Sobolev spaces, which analytically interesting it...

For $q>2, \gamma > 1$, we prove that maximal regularity of $L^q$ type holds for periodic solutions to $-\Delta u + |Du|^\gamma = f$ in $\mathbb{R}^d$, under the (sharp) assumption $q d \frac{\gamma-1}\gamma$.