First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalisation of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's |E|-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979: however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.

Transcendental number theory, by Alan Baker. Pp. x, 147. £4·90. 1975. SBN 0 521 20461 5 (Cambridge University Press)

The Geometry of the Zeros of a Polynomial in a complex variable. By M. Marden. Pp. ix, 183. $5. 1949. Mathematical Surveys, 3. (American Mathematical Society)

We obtain the Binet’s formula for k-Pell numbers and as a consequence we get some properties for k-Pell numbers. Also we give the generating function for k-Pell sequences and another expression for the general term of the sequence, using the ordinary generating function, is provided. Mathematics Subject Classification: 11B37, 05A15, 11B83.

On some identities and generating functions for k- Pell numbers

The fibonacci numbers

We obtain the Binet’s formula for k-Pell-Lucas numbers and as a consequence we obtain some properties for k-Pell-Lucas numbers. Also we give the generating function for the k-Pell-Lucas sequences and another expression for the general term of the sequence, using the ordinary generating function, is provided. Mathematics Subject Classification: 11B37, 05A15, 11B83.

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On Some Identities and Generating Functions for k-Pell-Lucas Sequence

Let F n be the nth Fibonacci number. The Fibonomial coefficients $\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F$ are defined for n ≥ k > 0 as follows 

$$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = \frac{{F_n F_{n - 1} \cdots F_{n - k + 1} }} {{F_1 F_2 \cdots F_k }},$$

with $\left[ {\begin{array}{*{20}c} n \\ 0 \\ \end{array} } \right]_F = 1$ and $\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = 0$. In this paper, we shall provide several identities among Fibonomial coefficients. In particular, we prove that 

$$\sum\limits_{j = 0}^{4l + 1} {\operatorname{sgn} (2l - j)\left[ {\begin{array}{*{20}c} {4l + 1} \\ j \\ \end{array} } \right]_F F_{n - j} = \frac{{F_{2l - 1} }} {{F_{4l + 1} }}\left[ {\begin{array}{*{20}c} {4l + 1} \\ {2l} \\ \end{array} } \right]_F F_{n - 4l - 1} ,}$$

holds for all non-negative integers n and l.

https://www.degruyter.com/downloadpdf/j/ms.2014.64.issue-4/s12175-014-0241-7/s12175-014-0241-7.xml

On some new identities for the Fibonomial coefficients

Let $ F_n$ be the $n$th Fibonacci number and let $L_n$ be the $n$th Lucas number. The order of appearance $z(n)$ of a natural number $n$ is defined as the smallest natural number $k$ such that $n$ divides $F_k$. For instance, $z(F_n)=n=z(L_n)/2$, for all $n>2$. In this paper, among other things, we prove that \begin{center} $z(L_{n}L_{n+1}L_{n+2}L_{n+3}L_{n+4})=\dfrac{n(n+1)(n+2)(n+3)(n+4)}{12}$, \end{center} for all positive integers $n\equiv 0,8\pmod{12}$.

/pdf/the-order-of-appearance-of-the-product-of-five-consecutive-2bo73w4zg3.pdf

The Order of Appearance of the Product of Five Consecutive Lucas Numbers

Let be the th Fibonacci number. The order of appearance of a natural number is defined as the smallest natural number such that divides . For instance, for all , we have . In this paper, we will construct infinitely many natural numbers satisfying the previous inequalities and which do not belong to the Fibonacci sequence.

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On Integer Numbers with Locally Smallest Order of Appearance in the Fibonacci Sequence

Perfect powers among Fibonomial coefficients

In 2000, Florian Luca proved that F10 = 55 and L5 = 11 are the largest numbers with only one distinct digit in the Fibonacci and Lucas sequence, respectively. In this paper, we find terms of a linear recurrence sequence with only one block of digits in its expansion in base g ≥ 2. As an application, we generalize Luca’s result by finding the Fibonacci and Lucas numbers with only one distinct block of digits of length up to 10 in its decimal expansion.

Diego Marques

Papers

On some new identities for the Fibonomial coefficients

The Order of Appearance of the Product of Five Consecutive Lucas Numbers

On Integer Numbers with Locally Smallest Order of Appearance in the Fibonacci Sequence

Perfect powers among Fibonomial coefficients

On terms of linear recurrence sequences with only one distinct block of digits