Abstract:
A nondecreasing sequence of positive integers is $(\alpha,\beta)$-Conolly, or Conolly-like for short, if for every positive integer $m$ the number of times that $m$ occurs in the sequence is $\alpha + \beta r_m$, where $r_m$ is $1$ plus the 2-adic valuation of $m$. A recurrence relation is $(\alpha, \beta)$-Conolly if it has an $(\alpha, \beta)$-Conolly solution sequence. We discover that Conolly-like sequences often appear as solutions to nested (or meta-Fibonacci) recurrence relations of the form $A(n) = \sum_{i=1}^k A(n-s_i-\sum_{j=1}^{p_i} A(n-a_{ij}))$ with appropriate initial conditions. For any fixed integers $k$ and $p_1,p_2,\ldots, p_k$ we prove that there are only finitely many pairs $(\alpha, \beta)$ for which $A(n)$ can be $(\alpha, \beta)$-Conolly. For the case where $\alpha =0$ and $\beta =1$, we provide a bijective proof using labelled infinite trees to show that, in addition to the original Conolly recurrence, the recurrence $H(n)=H(n-H(n-2)) + H(n-3-H(n-5))$ also has the Conolly sequence as a solution. When $k=2$ and $p_1=p_2$, we construct an example of an $(\alpha,\beta)$-Conolly recursion for every possible ($\alpha,\beta)$ pair, thereby providing the first examples of nested recursions with $p_i>1$ whose solutions are completely understood. Finally, in the case where $k=2$ and $p_1=p_2$, we provide an if and only if condition for a given nested recurrence $A(n)$ to be $(\alpha,0)$-Conolly by proving a very general ceiling function identity.

Abstract:
We explore a family of nested recurrence relations with arbitrary levels of nesting, which have an interpretation in terms of fixed points of morphisms over a countably infinite alphabet. Recurrences in this family are related to a number of well-known sequences, including Hofstadter's G sequence and the Conolly and Tanny sequences. For a recurrence a(n) in this family with only finitely terms, we provide necessary and sufficient conditions for the limit a(n)/n to exist.

Abstract:
Using a simple recurrence relation we give a new method to compute Jones polynomials of closed braids: we find a general expansion formula and a rational generating function for Jones polynomials. The method is used to estimate degree of Jones polynomials for some families of braids and to obtain general qualitative results.

Abstract:
In this paper we propose a definition of a recurrence relation homomorphism and illustrate our definition with a few examples. We then define the period of a k-th order of linear recurrence relation and deduce certain preliminary results associated with them.

Abstract:
Combinatorial interpretation of the fibonomial coefficients recently proposed by the present author results here in combinatorial interpretation of the recurrence relation for fibonomial coefficients . The presentation is provided with quite an exhaustive context where apart from plane grid coordinate system used several figures illustrate the exposition of statements and the derivation of the recurrence itself.

Abstract:
Recurrence formulae for arbitrary hydrogenic radial matrix elements are obtained in the Dirac form of relativistic quantum mechanics. Our approach is inspired on the relativistic extension of the second hypervirial method that has been succesfully employed to deduce an analogous relationship in non relativistic quantum mechanics. We obtain first the relativistic extension of the second hypervirial and then the relativistic recurrence relation. Furthermore, we use such relation to deduce relativistic versions of the Pasternack-Sternheimer rule and of the virial theorem.

Abstract:
Turning the skein relation for HOMFLY into a Fibonacci recurrence, we prove that there are only three rational specializations of HOMFLY polynomial: Alexander-Conway, Jones, and a new one. Using the recurrence relation, we find general and relative expansion formulae and rational generating functions for Alexander-Conway polynomial and the new polynomial, which reduce the computations to closure of simple braids, a subset of square free braids; HOMFLY polynomials of these simple braids are also computed. Algebraic independence of these three polynomials is proved.

Abstract:
In this note, starting with a little-known result of Kuo, I derive a recurrence relation for the Bernoulli numbers $B_{2 n}$, $n$ being any positive integer. This new recurrence seems advantageous in comparison to other known formulae since it allows the computation of both $B_{4 n}$ and $B_{4 n +2}$ from only $B_0, B_2,..., B_{2n}$.

Abstract:
We derive the recurrence relation of irreducible tensor operator for O(4) in using the Wigner-Eckart theorem. The physical process like radiative transitions in atomic physics, nuclear transitions between excited nuclear states can be described by the matrix element of an irreducible tensor, which is expressible in terms of a sum of products of two factors, one is a symmetry-related geometric factor, the Clebsch-Gordan coefficients, and the other is a physical factor, the reduced matrix elements. The specific properties of the states enter the physical factor only. It is precisely this fact that makes the Wigner-Eckart theorem invaluable in physics. Often time one is interested in ratio of two transition matrix element where it is sufficient to regard only the Clebsch-Gordan coefficients. In this paper we first get the commutation relations of O(4), and then we choose one of these relations to operate on the certain eigenvectors. Finally we take summation over all possible eigenvectors and obtain rather the compact recurrence relation for irreducible tensor operators.