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Register a new userA partial theta function Borwein conjecture
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We present an infinite family of Borwein type $+ - - $ conjectures. The expressions in the conjecture are related to multiple basic hypergeometric series with Macdonald polynomial argument.
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In the context of the (generalized) Delta Conjecture and its compositional form, DAdderio, Iraci, and Wyngaerd recently stated a conjecture relating two symmetric function operators, $D_k$ and $Theta_k$. We prove this Theta Operator Conjecture, finding it as a consequence of the five-term relation of Mellit and Garsia. As a result, we find surprising ways of writing the $D_k$ operators.
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In this paper we set up a bivariate representation of partial theta functions which not only unifies some famous identities for partial theta functions due to Andrews and Warnaar, et al. but also unveils a new characteristic of such identities. As further applications, we establish a general form of Warnaars identity and a general $q$--series transformation associated with Bailey pairs via the use of the power series expansion of partial theta functions.
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The Markoff injectivity conjecture states that $wmapstomu(w)_{12}$ is injective on the set of Christoffel words where $mu:{mathtt{0},mathtt{1}}^*tomathrm{SL}_2(mathbb{Z})$ is a certain homomorphism and $M_{12}$ is the entry above the diagonal of a $2times2$ matrix $M$. Recently, Leclere and Morier-Genoud (2021) proposed a $q$-analog $mu_q$ of $mu$ such that $mu_{qto1}(w)=mu(w)$ is the Markoff number associated to the Christoffel word $w$. We show that for every $q>0$, the map ${mathtt{0},mathtt{1}}^*tomathbb{Z}[q]$ defined by $wmapstomu_q(w)_{12}$ is injective over the language $mathcal{L}(s)$ of a balanced sequence $sin{mathtt{0},mathtt{1}}^mathbb{Z}$. The proof is based on new equivalent definitions of balanced sequences.
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