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In this paper we study sign changes of an infinite class of $\eta$-quotients which are holomorphic modular forms. There is also a relation to Hurwitz class numbers.

In this paper, we study sign changes of weakly holomorphic modular forms which are given as $\eta$-quotients. We give representative examples for forms of negative weight, weight zero, and positive weight.

We put forward several general conjectures concerning the algebraicity or transcendence of continued fractions and Stieltjes continued fractions defined by the Thue-Morse and period-doubling sequences in characteristic $2$. We present our Guess'n'Prove method, in which we exploit the structure of automata, for proving some of our conjectures in special cases.

Continued fraction expansions and Hankel determinants of automatic sequences are extensively studied during the last two decades. These studies found applications in number theory in evaluating irrationality exponents. The present paper is motivated by the converse problem: to study continued fractions of which the elements form an automatic sequence. We consider two such continued fractions defined by the Thue-Morse and period-doubling sequences respectively, and prove that they are congruent to algebraic series in $\mathbb{Z}[[x]]$ modulo $4$. Consequently, the sequences of the coefficients of the power series expansions of the two continued fractions modulo $4$ are $2$-automatic. Our approach is to first guess the explicit formulas of certain subsequences of $(P_n(x))$ and $(Q_n(x))$, where $P_n(x)/Q_n(x)$ is the canonical representation of the truncated continued fractions, then prove these formulas by an intricate induction involving eight subsequences while exploiting the relations between these subsequences.

Hankel determinants and automatic sequences are two classical subjects widely studied in Mathematics and Theoretical Computer Science. However, these two topics were considered totally independently, until in 1998, when Allouche, Peyri\`ere, Wen and Wen proved that all the Hankel determinants of the Thue-Morse sequence are nonzero. This property allowed Bugeaud to prove that the irrationality exponents of the Thue-Morse-Mahler numbers are exactly 2. Since then, the Hankel determinants of several other automatic sequences, in particular, the paperfolding sequence, the Stern sequence, the period-doubling sequence, are studied by Coons, Vrbik, Guo, Wu, Wen, Bugeaud, Fu, Han, Fokkink, Kraaikamp, and Shallit. On the other hand, it is known that the Hankel determinants of a rational power series are ultimately zero, and the Hankel determinants of a quadratic power series over finite fields are ultimately periodic. It is therefore natural to ask if we can obtain similar results about the Hankel determinants of algebraic series. In the present paper, we provide a partial answer to this question by establishing the automaticity of the reduced Hankel determinants modulo $2$ of $\pm 1$-automatic sequences with kernel of cardinality at most $2$.

The classical Eulerian polynomials can be expanded in the basis $t^{k-1}(1+t)^{n+1-2k}$ ($1\leq k\leq\lfloor (n+1)/2\rfloor$) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian polynomials. In this paper, we prove a $q$-analogue of this expansion for Carlitz's $q$-Eulerian polynomials as well as a similar formula for Chow-Gessel's $q$-Eulerian polynomials of type $B$. We shall give some applications of these two formulae, which involve two new sequences of polynomials in the variable $q$ with positive integral coefficients. An open problem is to give a combinatorial interpretation for these polynomials., Comment: 13 pages, to appear in The Ramanujan Journal

We find a $q$-analog of the following symmetrical identity involving binomial coefficients $\binom{n}{m}$ and Eulerian numbers $A_{n,m}$, due to Chung, Graham and Knuth [{\it J. Comb.}, {\bf 1} (2010), 29--38]: {equation*} \sum_{k\geq 0}\binom{a+b}{k}A_{k,a-1}=\sum_{k\geq 0}\binom{a+b}{k}A_{k,b-1}. {equation*} We give two proofs, using generating function and bijections, respectively., Comment: S\'eminaire Lotharingien de Combinatoire, vol. 67(2012), Article B67c11 pages