The golden ratio, Fibonacci numbers and BBP-type formulas

We derive interesting arctangent identities involving the golden ratio, Fibonacci numbers and Lucas numbers. Binary BBP-type formulas for the arctangents of certain odd powers of the golden ratio are also derived, for the first time in the literature. Finally we derive golden-ratio-base BBP-type formulas for some mathematical constants, including $\pi$, $\log 2$, $\log\phi$ and $\sqrt 2\,\arctan\sqrt 2$. The $\phi-$nary BBP-type formulas derived here are considerably simpler than similar results contained in earlier literature

Partial sums and generating functions for powers of second order sequences with indices in arithmetic progression

The sums $\sum_{j = 0}^k {u_{rj + s}^{2n}z^j }$, $\sum_{j = 0}^k {u_{rj + s}^{2n-1}z^j }$, $\sum_{j = 0}^k {v_{rj + s}^{n}z^j }$ and $\sum_{j = 0}^k {w_{rj + s}^{n}z^j }$ are evaluated; where $n$ is any positive integer, $r$, $s$ and $k$ are any arbitrary integers, $z$ is arbitrary, $(u_i)$ and $(v_i)$ are the Lucas sequences of the first kind, and of the second kind, respectively; and $(w_i)$ is the Horadam sequence. Pantelimon St\uanic\ua set out to evaluate the sum $\sum_{j = 0}^k {w_j^n z^j }$. His solution is not complete because he made the assumption that $w_0=0$, thereby giving effectively only the partial sum for $(u_i)$, the Lucas sequence of the first kind.Comment: 8 pages, no figures, no table

A master identity for Horadam numbers

We derive an identity involving Horadam numbers. Numerous new identities as well as those found in the existing literature are subsumed in this single identity.Comment: 14 pages, no figures, no table

A novel approach to the discovery of binary BBP-type formulas for polylogarithm constants

Using a clear and straightforward approach, we discover and prove new binary digit extraction BBP-type formulas for polylogarithm constants. Some known results are also rediscovered in a more direct and elegant manner. Numerous experimentally discovered and previously unproved binary BBP-type formulas are also proved

Infinite arctangent sums involving Fibonacci and Lucas numbers

Using a straightforward elementary approach, we derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of celebrated results appear as particular cases of the more general formulas derived here

Some remarkable infinite product identities involving Fibonacci and Lucas numbers

By applying the classic telescoping summation formula and its variants to identities involving inverse hyperbolic tangent functions having inverse powers of the golden ratio as arguments and employing subtle properties of the Fibonacci and Lucas numbers, we derive interesting general infinite product identities involving these numbers.Comment: 11 pages, corrected a typ

A new Fibonacci identity and its associated summation identities

We derive a new Fibonacci identity. This single identity subsumes important known identities such as those of Catalan, Ruggles, Halton and others, as well as standard general identities found in the books by Vajda, Koshy and others. We also derive several binomial and ordinary summation identities arising from this identity; in particular we obtain a generalization of Halton's general Fibonacci identity.Comment: 12 pages, no figure

A novel approach to the discovery of ternary BBP-type formulas for polylogarithm constants

Using a clear and straightforward approach, we prove new ternary (base 3) digit extraction BBP-type formulas for polylogarithm constants. Some known results are also rediscovered in a more direct and elegant manner. A previously unproved degree~4 ternary formula is also proved. Finally, a couple of ternary zero relations are established, which prove two known but hitherto unproved formulas

A non-PSLQ route to BBP-type formulas

BBP-type formulas are usually discovered experimentally, through computer searches. In this paper, however, starting with two simple generators, and hence without doing any computer searches, we derive a wide range of BBP-type formulas in general bases. Many previously discovered BBP-type formulas turn out to be particular cases of the formulas derived here

Summation Identities Involving Padovan and Perrin Numbers

Unlike in the case of Fibonacci and Lucas numbers, there is a paucity of literature dealing with summation identities involving the Padovan and Perrin numbers. In this paper, we derive various summation identities for these numbers, including binomial and double binomial identities. Our results derive from the rich algebraic properties exhibited by the zeros of the characteristic polynomial of the Padovan/Perrin sequence.Comment: 18 pages, no figures, made a typographical correctio