The main motivation of this paper is to investigate some derivative properties of the generating functions for the numbers Yn(λ) and the polynomials Yn(x;λ), which were recently introduced by Simsek ...30. We give functional equations and differential equations (PDEs) of these generating functions. By using these functional and differential equations, we derive not only recurrence relations, but also several other identities and relations for these numbers and polynomials. Our identities include the Apostol–Bernoulli numbers, the Apostol–Euler numbers, the Stirling numbers of the first kind, the Cauchy numbers and the Hurwitz–Lerch zeta functions. Moreover, we give hypergeometric function representation for an integral involving these numbers and polynomials. Finally, we give infinite series representations of the numbers Yn(λ), the Changhee numbers, the Daehee numbers, the Lucas numbers and the Humbert polynomials.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In this paper, we mainly prove the following conjectures of Sun 16: Let p > 3 be a prime. Then
\begin{align*}
&A_{2p}\equiv A_2-\frac{1648}3p^3B_{p-3}\ ({\rm{mod}}\ p^4),\\
&A_{2p-1}\equiv ...A_1+\frac{16p^3}3B_{p-3}\ ({\rm{mod}}\ p^4),\\
&A_{3p}\equiv A_3-36738p^3B_{p-3}\ ({\rm{mod}}\ p^4),
\end{align*} where $A_n=\sum_{k=0}^n\binom{n}k^2\binom{n+k}{k}^2$ is the nth Apéry number, and Bn is the nth Bernoulli number.
The aim of this paper is to obtain some interesting infinite series representations for the Apostol-type parametrically generalized polynomials with the aid of the Laplace transform and generating ...functions. In particular, by using the method of generating functions, we derive not only recurrence relations, but also several other formulas, identities, and relations as well as combinatorial sums for these parametrically generalized numbers and polynomials and for other known special numbers and polynomials. These identities, relations and combinatorial sums are related to the two-parameter types of the Apostol–Bernoulli polynomials of higher order, the two-parameter types of Apostol–Euler polynomials of higher order, the two-parameter types of Apostol–Genocchi polynomials of higher order, the Apostol–Bernoulli polynomials of higher order, the Apostol–Euler polynomials of higher order, the Apostol–Genocchi polynomials of higher order, the cosine- and sine-Bernoulli polynomials, the cosine- and sine-Euler polynomials, the
λ
-array-type polynomials, the
λ
-Stirling numbers, the polynomials
C
n
x
,
y
, and the polynomials
S
n
x
,
y
. Finally, we present several new recurrence relations for these special polynomials and numbers.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
45.
A mental number line in human newborns Di Giorgio, Elisa; Lunghi, Marco; Rugani, Rosa ...
Developmental science,
November 2019, 2019-11-00, 20191101, Volume:
22, Issue:
6
Journal Article
Open access
Humans represent numbers on a mental number line with smaller numbers on the left and larger numbers on the right side. A left‐to‐right oriented spatial–numerical association, (SNA), has been ...demonstrated in animals and infants. However, the possibility that SNA is learnt by early exposure to caregivers’ directional biases is still open. We conducted two experiments: in Experiment 1, we tested whether SNA is present at birth and in Experiment 2, we studied whether it depends on the relative rather than the absolute magnitude of numerousness. Fifty‐five‐hour‐old newborns, once habituated to a number (12), spontaneously associated a smaller number (4) with the left and a larger number (36) with the right side (Experiment 1). SNA in neonates is not absolute but relative. The same number (12) was associated with the left side rather than the right side whenever the previously experienced number was larger (36) rather than smaller (4) (Experiment 2). Control on continuous physical variables showed that the effect is specific of discrete magnitudes. These results constitute strong evidence that in our species SNA originates from pre‐linguistic and biological precursors in the brain.
Neonates, after being habituated to a certain number of dots, associated a smaller number with the left and a larger number with the right side. This evidence demonstrates that neonates spontaneously associate numbers with space.
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DOBA, FZAB, GIS, IJS, IZUM, KILJ, NLZOH, NUK, OILJ, PILJ, PNG, SAZU, SBCE, SBMB, UILJ, UKNU, UL, UM, UPUK
The LHCb collaboration has recently observed three pentaquark peaks, the Pc(4312), Pc(4440) and Pc(4457). They are very close to a pair of heavy baryon-meson thresholds, with the Pc(4312) located 8.9 ...MeV below the DΣc threshold, and the Pc(4440) and Pc(4457) located 21.8 and 4.8 MeV below the D∗Σc one. The spin-parities of these three states have not been measured yet. In this work we assume that the Pc(4312) is a JP = 1/2− ¯DΣc bound state, while the Pc(4440) and Pc(4457) are ¯D∗Σc bound states of unknown spin-parity, where we notice that the consistent description of the three pentaquarks in the one-boson-exchange model can indeed determine the spin and parities of the later, i.e., of the two ¯ D∗Σc molecular candidates. For this determination we revisit first the one-boson-exchange model, which in its original formulation contains a short-range deltalike contribution in the spin-spin component of the potential. We argue that it is better to remove these deltalike contributions because, in this way, the one-boson-exchange potential will comply with the naïve expectation that the form factors should not have a significant impact in the long-range part of the potential (in particular the one-pion-exchange part). Once this is done, we find that it is possible to consistently describe the three pentaquarks, to the point that the Pc(4440) and Pc(4457) can be predicted from the Pc(4312) within a couple of MeV with respect to their experimental location. In addition the so-constructed one-boson-exchange model predicts the preferred quantum numbers of the Pc(4440) and Pc(4457) molecular pentaquarks to be 3/2− and 1/2−, respectively.
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CMK, CTK, FMFMET, IJS, NUK, PNG, UM
Inversion sequences have intriguing applications in combinatorics, computer sciences and polyhedral geometry. Ascent sequences, as one of the most important subsets of inversion sequences, were ...introduced by Bousquet-Mélou, Claesson, Dukes and Kitaev to encode the (2+2)-free posets. Pattern avoidance in ascent sequences was first studied by Ducan and Steingrímsson in 2011, while the systematic study of patterns in inversion sequences was initiated only recently by Corteel–Martinez–Savage–Weselcouch and Mansour–Shattuck. In this paper, we investigate systematically the enumeration of inversion or ascent sequences avoiding vincular patterns of length 3, where two of the three letters are required to be adjacent. Our results connect restricted inversion sequences and ascent sequences to a number of well-known combinatorial sequences including Bell numbers, Fishburn numbers, Powered Catalan numbers, Semi-Baxter numbers and the number of 3-noncrossing partitions. A variety of combinatorial bijections, known or newly constructed, are applied to achieve these interesting connections.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
An important issue in understanding mathematical cognition involves the similarities and differences between the magnitude representations associated with various types of rational numbers. For ...single-digit integers, evidence indicates that magnitudes are represented as analog values on a mental number line, such that magnitude comparisons are made more quickly and accurately as the numerical distance between numbers increases (the distance effect). Evidence concerning a distance effect for compositional numbers (e.g., multidigit whole numbers, fractions and decimals) is mixed. We compared the patterns of response times and errors for college students in magnitude comparison tasks across closely matched sets of rational numbers (e.g., 22/37, 0.595, 595). In Experiment 1, a distance effect was found for both fractions and decimals, but response times were dramatically slower for fractions than for decimals. Experiments 2 and 3 compared performance across fractions, decimals, and 3-digit integers. Response patterns for decimals and integers were extremely similar but, as in Experiment 1, magnitude comparisons based on fractions were dramatically slower, even when the decimals varied in precision (i.e., number of place digits) and could not be compared in the same way as multidigit integers (Experiment 3). Our findings indicate that comparisons of all three types of numbers exhibit a distance effect, but that processing often involves strategic focus on components of numbers. Fractions impose an especially high processing burden due to their bipartite (a/b) structure. In contrast to the other number types, the magnitude values associated with fractions appear to be less precise, and more dependent on explicit calculation.
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CEKLJ, FFLJ, NUK, ODKLJ, PEFLJ, UPUK
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