arXiv subjects
Publications and source records attributed to Mong Lung Lang.
We prove that $Γ_0(2^n)$ ($n\ge2$) has a Farey sequence $\{e_i\}$ such that $e_i \le 2^{n-1}$ for all $e_i$. The above upper bound is optimal, and there exists a unique $j$ such that $e_j= 2^{n-1} $. For each $e_i$, there exists a unique $a_i$ such that $\{ a_i/e_i\}\cup \{\infty\}$ is the set of ideal vertices of a fundamental domain of $Γ_0(2^n)$ whose side-pairings give a set of independent generators of $Γ_0(2^n)$.
Let $n$ be a prime or its square. We prove that the congruence subgroup $Γ_0(n)$ admits a free product decomposition into cyclic factors in such a way that the $(2,1)$-component of each cyclic generator is either $n$ or $0$, answering a conjecture of Kulkarni. We can also require that the Frobenius norm of each generator is less than $2n-1$. A crucial observation is that if $P$ denotes the convex hull of the extended Farey sequence of order $\lfloor \sqrt{n} \rfloor$ in the hyperbolic plane $\mathbb{H}^2$, then the projection $π: \mathbb{H}^2\to \mathbb{H}^2/Γ_0(n)$ is injective on the interior of $P$ and each connected component of $π(\mathbb{H}^2)\setminusπ(P)$ is either an order-three cone of area $π/3$ or an ideal triangle. Denoting by $m(Γ_0(n))$ the minimum of the largest denominator in the cusp set of $Q$ where $Q$ ranges over all possible special (fundamental) polygons for $Γ_0(n)$, we establish the inequality $ \lfloor \sqrt{n} \rfloor \le m(Γ_0(n))\le \lfloor \sqrt{4n/3} \rfloor$, and completely characterize the cases in which the bounds are achieved. We also prove analogous results when $n$ is the multiplication of two sufficiently close odd primes.
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 $F_3(w, x)$.
Following Bridgeman, we demonstrate several families of infinite dilogarithm identities associated with Fibonacci numbers, Lucas numbers, convergents of continued fractions of even periods, and terms arising from various recurrence relations.
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.