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In this note I collect some typical examples of orthogonal polynomials with simple moments where both moments and orthogonal polynomials have nice q-analogs.

We present a proof of a conjecture of Goh and Wildberger on the factorization of the spread polynomials. We indicate how the factors can be effectively calculated and exhibit a connection to the factorization of Fibonacci numbers into primitive parts., Comment: 8 pages

We obtain formulas for the coefficients of positive and negative powers of a partial theta function., Comment: 5 pages

We study some polynomials which are related to Hankel determinants of backward shifts of the coefficients of a partial theta function. In this version an appendix is added which gives a simple formula for the coefficients of the reciprocal of the partial theta function., Comment: Change of title, appendix added

This note presents some results about Hankel determinants of backwards shifted Catalan-like numbers and related sequences., Comment: slight extension of previous version

Using a slightly generalized result of George Andrews and Jet Wimp this note gives a simple computational proof of some Hankel determinants of backwards shifts of convolution powers of Catalan numbers and obtains analogous results for Narayana polynomials., Comment: 11 pages

Martin Aigner introduced Catalan-like numbers as elements of the first column of admissible matrices and studied Hankel determinants of their forward shifts. In this paper we collect some properties of the Hankel determinants of the other columns which are suggested by computer experiments. By prepending zero rows to admissible matrices we also consider Hankel determinants of backward shifts., Comment: 14 pages

Computer experiments suggest some conjectures about Hankel determinants of convolution powers of Catalan numbers. Unfortunately, for most of them I have no proofs. I would like to present them anyway hoping that someone finds them interesting and can prove them., Comment: 12 pages, a new paragraph has been added

By prepending zeros to a given sequence Hankel determinants of backward shifts of this sequence become meaningful. We obtain some results for the sequences of Catalan numbers and of some numbers and polynomials which are related to Catalan numbers and propose conjectures for sequences of convolution powers of Catalan numbers., Comment: 14 pages

We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials., Comment: 15 pages

Based on well-known properties of Fibonacci and Lucas numbers and polynomials we give a self-contained approach to some bivariate analogs., Comment: 21 pages

This note collects some results and conjectures for the generating functions of the Hankel determinants of certain polynomials which are related to Motzkin paths., Comment: An Addendum has been added

Temperature is one of the most relevant parameters for the regulation of intracellular processes. Measuring localized subcellular temperature gradients is fundamental for a deeper understanding of cell function, such as the genesis of action potentials, and cell metabolism. Here, we detect for the first time temperature variations (1{\deg}C) associated with potentiation and depletion of neuronal firing, exploiting a nanoscale thermometer based on optically detected magnetic resonance in nanodiamonds. Our results provide a tool for assessing neuronal spiking activity under physiological and pathological conditions and, conjugated with the high sensitivity of this technique (in perspective sensitive to < 0.1{\deg}C variations), pave the way to a systematic study of the generation of localized temperature gradients. Furthermore, they prompt further studies explaining in detail the physiological mechanism originating this effect., Comment: 27 pages, 5 figures, 3 tables

This note collects some facts and conjectures about the Hankel determinants and their generating functions of the columns of Hoggatt triangles which apparently are related to combinatorial objects such as Young tableaux and Narayana numbers., Comment: Slightly extended version with an additional reference

The middle binomial coefficients can be interpreted as numbers of Motzkin paths which have no horizontal steps at positive heights. Assigning suitable weights gives some nice polynomial extensions. We determine the Hankel determinants and their generating functions for the middle binomial coefficients and derive many conjectures for their polynomial extensions. Finally, we explore experimentally some modifications of the middle binomial coefficients whose Hankel determinants show an interesting modular pattern and obtain some q-analogs., Comment: some typos corrected

In this (partly expository) paper we give a short overview about the close relationship between the sequence of Catalan numbers and Hankel determinants from the point of view of orthogonal polynomials and show that an analogous situation exists for more general sequences., Comment: 18 pages

We give an overview about some elementary properties of Hoggatt matrices, which are generalizations of Pascal triangle, and study q-analogs and Fibonacci analogs and derive a common generalization., Comment: 28 pages, some corrections and simplifications

Diamond nanoparticles (nanodiamonds) can transport active drugs in cultured cells as well as in vivo. However, in the latter case, methods allowing to determine their bioavailability accurately are still lacking. Nanodiamond can be made fluorescent with a perfectly stable emission and a lifetime ten times longer than the one of tissue autofluorescence. Taking advantage of these properties, we present an automated quantification method of fluorescent nanodiamonds (FND) in histological sections of mouse organs and tumor, after systemic injection. We use a home-made time-delayed fluorescence microscope comprising a custom pulsed laser source synchronized on the master clock of a gated intensified array detector. This setup allows to obtain ultra-high-resolution images 120 Mpixels of whole mouse organs sections, with subcellular resolution and single-particle sensitivity. As a proof-of-principle experiment, we quantified the biodistribution and aggregation state of new cationic FNDs able to transport small interfering RNA inhibiting the oncogene responsible for Ewing sarcoma. Image analysis showed a low yield of nanodiamonds in the tumor after intravenous injection. Thus, for the in vivo efficacy assay we injected the nanomedicine into the tumor. We achieved a 28-fold inhibition of the oncogene. This method can readily be applied to other nanoemitters with $\approx$100 ns lifetime., Comment: Nanoscale, Royal Society of Chemistry, 2021

The theme of this article is a "reciprocity" between bounded up-down paths and bounded alternating sequences. Roughly speaking, this ``reciprocity" manifests itself by the fact that the extension of the sequence of numbers of paths of length $n$, consisting of diagonal up- and down-steps and being confined to a strip of bounded width, to negative $n$ produces numbers of alternating sequences of integers that are bounded from below and from above. We show that this reciprocity extends to families of non-intersecting bounded up-down paths and certain arrays of alternating sequences which we call alternating tableaux. We provide as well weighted versions of these results. Our proofs are based on Viennot's theory of heaps of pieces and on the combinatorics of non-intersecting lattice paths. An unexpected application leads to a refinement of a result of Bousquet-M\'elou and Viennot on the width-height-area generating function of parallelogram polyominoes. Finally, we exhibit the relation of the arising alternating tableaux to plane partitions of strip shapes., Comment: 77 pages, AmS-TeX; added a general reciprocity theorem, Theorem 40; numerous improvements of presentation

Biocompatible nanoscale probes for sensitive detection of paramagnetic species and molecules associated with their (bio)chemical transformations would provide a desirable tool for a better understanding of cellular redox processes. Here, we describe an analytical tool based on quantum sensing techniques. We magnetically coupled negatively charged nitrogen-vacancy (NV) centers in nanodiamonds (NDs) with nitroxide radicals present in a bioinert polymer coating of the NDs. We demonstrated that the T1 spin relaxation time of NV centers is very sensitive to the number of nitroxide radicals, with a resolution down to ~10 spins per ND (detection of approximately $ 10^-23 $ mol in a localized volume). The detection is based on T1 shortening upon the radical attachment and we propose a theoretical model describing this phenomenon. We further show this colloidally stable, water-soluble system can be used dynamically for spatiotemporal readout of a redox chemical process (oxidation of ascorbic acid) occurring near the ND surface in an aqueous environment under ambient conditions., Comment: 26 pages (plus 18 pages of supporting information), 3 figures (plus 8 figures in supporting information)

We give an overview about the power product expansion of the exponential series and derive some q-analogs

We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalising known results for Catalan numbers., Comment: 28 pages, AmS-TeX; some typos corrected

Starting with some determinants of binomial coefficients which are related to Fibonacci and Lucas polynomials we study similar determinants for some generalizations of these polynomials and their q-analogues., Comment: 22 pages

In this (mostly expository) paper I want to share some observations prompted by a class of matrices whose determinants are Catalan numbers. Considering different methods of proof we obtain some generalizations and q-analogues and connections with Hankel determinants. Finally we state some conjectures., Comment: 46 pages

For small $r$ the Hankel determinants $d_r(n)$ of the sequence $\left({2n+r\choose n}\right)_{n\ge 0}$ are easy to guess and show an interesting modular pattern. For arbitrary $r$ and $n$ no closed formulae are known, but for each positive integer $r$ the special values $d_r(rn)$, $d_r(rn+1)$, and $d_r(rn+\lfloor\frac{r+1}{2}\rfloor)$ have nice values which will be proved in this paper., Comment: 20 pages

We consider Hankel determinants of the sequence of Catalan numbers modulo 2 (interpreted as integers 0 and 1) and more generally Hankel determinants where the sum over all permutations reduces to a single signed permutation., Comment: 34 pages

This (partly expository) paper originated from the study of Hankel determinants of convolution powers of Catalan numbers and of Narayana polynomials. This led to some Hankel determinants of signed Catalan numbers whose values are multiples of Fibonacci numbers and to some Hankel determinants of signed central binomial coefficients whose values are multiples of Lucas numbers. Most proofs are computational but we also include a combinatorial one due to Christian Krattenthaler. Finally we formulate some conjectures., Comment: 31 pages, 2 figures

In this expository paper we collect some simple facts about analogues of Pascals triangle where the entries count subsets of the integers with an even or odd sum and show that they are related to Rogers-Szego polynomials. In particular we consider an interesting triangle due to Losanitsch from this point of view. We also sketch some extensions of these results to sets whose sums have fixed residues modulo a prime $p$., Comment: 23 pages

We give an overview of known results about Hilbert matrices from the point of view of orthogonal polynomials and compute Hankel determinants of harmonic numbers and related topics., Comment: 18 pages

We give an elementary account of generalized Fibonacci and Lucas polynomials whose moments are Narayana polynomials of type A and type B., Comment: 30 pages

We show that q-Catalan numbers, q- central binomial coefficients and q- Narayana polynomials are moments of q-analogues of Fibonacci and Lucas polynomials and related polynomials., Comment: 15 pages

We give a survey of some known and some new results about factors of different sorts of $q-$Fibonacci numbers., Comment: Since I became aware of the fact that some of my experimentally found results have already been proved in a paper by H. Pan I changed the title and revised the paper thoroughly

The Rogers-Szeg\"o polynomials are natural q-analogues of Newton binomials. In general they have no closed expression. We consider some exceptional cases which are products of a factor with a closed formula and another one with nice values for q=1 and q=-1. These are related to normalized Rogers-Szeg\"o polynomials which have nice expansions and Hankel determinants. The Exposition of the paper is elementary and almost self-contained., Comment: Enlarged and updated Version

Situational Judgement Questions (SJQs) measuring process quality were included in the OECD Starting Strong Teaching and Learning International Survey 2018 (TALIS Starting Strong 2018) to address concerns of self-report bias in large-scale international surveys. These SJQs provide the staff in early childhood education and care with situations taken from a real-life professional context and offer several options on how to address these given situations. Using TALIS Starting Strong 2018 field trial data, this paper evaluates the reliability and validity of the SJQs as measures of process quality in a large-scale international survey. The results showed that the SJQs were reliable, valid and worked well in evaluating process quality. High process quality was characterised by: 1) supporting child-directed play; 2) managing conflicts through behavioural management; and 3) supporting pro-social behaviour by encouraging sharing and collaboration among children. Staff self efficacy and formal education were positively related to these practices. The paper further makes recommendations regarding the formats, type of analysis and interpretation of the SJQs in the Main Survey.

In the first part of this paper I give an elementary overview about some number sequences which count various sorts of lattice paths in strips along the x-axis and compute their generating functions in terms of Fibonacci and Lucas polynomials. In the second part I generalize these results by introducing suitable weights and study some special cases in more detail. In the course of this work I have been led to curious number triangles and various conjectures, Comment: Simplified and updated version

In this note we give a survey about polynomials whose moments are multiples of super Catalan numbers and explore two different kinds of q-analogues.