In this paper, we evaluate determinants of some families of Toeplitz{Hessenberg matrices having TRIBONACCI number entries. These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and TRIBONACCI numbers. In particular, we establish a connection between the TRIBONACCI and the Fibonacci and Padovan sequences via Toeplitz{Hessenberg determinants. We then obtain, by combinatorial arguments, extensions of our determinant formulas in terms of generalized TRIBONACCI sequences satisfying a recurrence of the form T(r) n = T(r) n-1 +T(r) n-2 +T(r) n-r for n  r, with the appropriate initial conditions, where r 3 is arbitrary.

At the Fourty-forth International Symposium on FUNCTIONAL EQUATIONs in 2006, Louisville, Kentucky, USA, T.M.K. Davison presented a talk under the title "D’Alembert’s FUNCTIONAL EQUATION on compact groups" in which he offered a solution method for this EQUATION on any compact group. In this paper we offer another approach based on some very recent results on spectral synthesis, which method seems to be useful in the case of other EQUATIONs.

Let $S$ be a semigroup. In the present paper, we determine the complex-valued solutions $(f,g)$ of the FUNCTIONAL EQUATION$$g(x\sigma (y)) = g(x)g(y)-f(x)f(y)+\alpha f(x\sigma(y)),\  x,y\in S,$$where $\sigma :S\rightarrow S$ is an automorphism that need not be involutive, and $\alpha \in \mathbb{C}$ is a fixed constant. Our results generalize and extend the ones by Stetk\ae r in The cosine addition law with an additional term. Aequat Math., no. 6, 90, 1147-1168 (2016), and also the ones by Aserrar and Elqorachi in A generalization of the cosine addition law on semigroups. Aequat Math. 97, 787–804 (2023). Some consequences of our results are presented.

Let $(G,+)$ be an abelian group and $Y$  a linear space over the field $\Bbb{F}\in\{\Bbb{R}, \Bbb{C}\}$. In this paper, we investigate the conditional Cauchy FUNCTIONAL EQUATION \[f(x+y)\neq af(x)+bf(y)\quad\Rightarrow \quad f(x+y)=f(x)+f(y),\quad x,y\in G,\] for functions $f:G\to Y$, where  $a,b\in \Bbb{F}$ are fixed constants. The general solution and stability of this FUNCTIONAL EQUATION are described.

In this paper we consider the general solution of a Jensen type FUNCTIONAL EQUATION. Moreover we prove the stability theorem of this EQUATION in the spirit of Hyers, Ulam, Rassias and Gâvruta.

In this paper, we introduce and solve a system of bi-Drygas FUNCTIONAL EQUATIONs \begin{EQUATION}\left\{\begin{aligned}        &f(x+y,z)+f(x-y,z)=2f(x,z)+f(y,z)+f(-y,-z)\nonumber\\        &f(x,y+z)+f(x, y-z)=2f(x,y)+f(x,z)+f(-x,-z)\nonumber\end{aligned}\right.\end{EQUATION}for all $x,y,z\in X$. We will also investigate the Hyers-Ulam stability of the system of bi-Drygas FUNCTIONAL EQUATIONs.