In this paper, we derive eight basic identities of symmetry in three
variables related to generalized Euler polynomials and alternating generalized
power sums. All of these are new, since there have been results only about
identities of symmetry in two variables. The derivations of identities are
based on the $p$-adic fermionic integral expression of the generating function
for the generalized Euler polynomials and the quotient of integrals that can be
expressed as the exponential generating function for the alternating
generalized power sums.

In this paper, we derive eight basic identities of symmetry in three
variables related to Euler polynomials and alternating power sums. These and
most of their corollaries are new, since there have been results only about
identities of symmetry in two variables. These abundance of symmetries shed new
light even on the existing identities so as to yield some further interesting
ones.

In this paper, we derive eight basic identities of symmetry in three
variables related to $q$-Bernoulli polynomials and the $q$-analogue of power
sums. These and most of their corollaries are new, since there have been
results only about identities of symmetry in two variables. These abundance of
symmetries shed new light even on the existing identities so as to yield some
further interesting ones.

In this paper, we construct a binary linear code connected with the
Kloosterman sum for $GL(2,q)$. Here $q$ is a power of two. Then we obtain a
recursive formula generating the power moments 2-dimensional Kloosterman sum,
equivalently that generating the even power moments of Kloosterman sum in terms
of the frequencies of weights in the code. This is done via Pless power moment
identity and by utilizing the explicit expression of the Kloosterman sum for
$GL(2,q)$.

In this paper, we construct eight infinite families of ternary linear codes
associated with double cosets with respect to certain maximal parabolic
subgroup of the special orthogonal group $SO^{-}(2n,q)$. Here ${q}$ is a power
of three. Then we obtain four infinite families of recursive formulas for power
moments of Kloosterman sums with square arguments and four infinite families of
recursive formulas for even power moments of those in terms of the frequencies
of weights in the codes.

In this paper, we construct two binary linear codes associated with
multi-dimensional and $m -$multiple power Kloosterman sums (for any fixed $m$)
over the finite field $\mathbb{F}_{q}$. Here $q$ is a power of two. The former
codes are dual to a subcode of the binary hyper-Kloosterman code. Then we
obtain two recursive formulas for the power moments of multi-dimensional
Kloosterman sums and for the $m$-multiple power moments of Kloosterman sums in
terms of the frequencies of weights in the respective codes.