arXiv subjects
Publications and source records attributed to Cheng Lien Lang.
A Fibonacci pair $F_s(w,x)$ of rank $s$ is a pair $s \times s$ nonsingular matrices such that $wx=xw$ and that the entries of $aw^n$ and $axw^m$ are polynomials of Fibonacci or Lucas numbers for some nonzero $a$. We construct identities systematically by the study of $F_2(w, x)$ and $F_3(w, x)$.
We study the arithmetic and geometry properties of the Hecke group $G_q$. In particular, we prove that $G_q$ has a subgroup $X $ of index $d$, genus $g$ with $v_{\infty} $ cusps, and $τ_2$ (resp. $v_{r_i}$) conjugacy classes of elements that are conjugates of $S$ (resp. $R^{q/r_i}$) if and only if (i) $ 2g-2 + τ_2/2 +\sum_{i=1}^k v_{r_i}(1-1/r_i) + v_{\infty} = d(1/2-1/q)$, and (ii) $ m _0= 4g-4 +τ_2 + 2 v_{\infty} + \sum _{i=1}^k v_{r_i}(2-q/r_i)\ge 0$ is a multiple of $q-2$, (iii) $m \ge 0$. In the case $q$ is odd, (ii) is a consequence of (i).
We give a simple algorithm that enables us to determine whether a subgroup of finite index of the Hecke group is normal.
The present article gives the index formula of the principal congruence subgroups of the Hecke group H_5
The present article determines the indices of the principal congruence subgroups of the Bianchi groups $B_d$, $SL(2,\mathcal O)$ and elementary matrix group $E$ and extends Wohlfahrt's Theorem to $B_d$, $SL(2,\mathcal O)$ and $E$, where $\mathcal O$ is the ring of integers of some number fields.
We prove that the power subgroup $G_5^5$ and the commutator subgroup $G_5'$ of the Hecke group $G_5$ are not congruence.
Let K be a subgroup of the inhomogeneous Hecke group G_5 of finite index. Suppose that the geometric level of K is r. Then K is congruence if and only if K contains the principal congruence subgroup of level 2r.
We study the three term recurrence modulo m. In particular, we prove that Pell numbers modulo m is residue complete if and only m is 2, a power of 3, or a power of 5. Pell-Lucas numbers modulo m is residue complete if and only if m is a power of 3.
We give necessary and sufficient conditions for a Fibonacci cycle to be residue complete (nondefective). In particular, the Lucas numbers modulo m is residue complete if and only if m = 2,4,6,7,14 or a power of 3.
We prove a stronger version of Jarden's Theorem for recurrence of powers of recursive functions
We study the Fibonacci and Lucas numbers and demonstrate how identities can be constructed by investigating trivalent graphs and splitting fields.
We study the existence of the identity L_n^2 -5F_n^2 = 4(-1)^n.
Let x(n) be a recurrence relation. The main purpose of this article is to determine a recurrence for powers of x(n).
We give alternative proof of Melham's and Howard's identities on generalised Fibonacci numbers
We study the recurrence of the product of n functions, each of which satisfies the same recurrence relation.
By investigating a recurrence relation about functions, we first give alternative proofs of various identities on Fibonacci numbers and Lucas numbers, and then, make certain well known identities visible via certain trivalent graph associated to the recurrence relation.