Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 01/2016, Volume 433, Issue 1, pp. 706 - 711

We prove the last remaining case of the original 13 Ramanujan-type supercongruence conjectures due to Van Hamme from 1997. The proof utilizes classical...

Wilf–Zeilberger pair | Ramanujan | Supercongruence | Wilf-Zeilberger pair | MATHEMATICS | MATHEMATICS, APPLIED | FORMULAS

Wilf–Zeilberger pair | Ramanujan | Supercongruence | Wilf-Zeilberger pair | MATHEMATICS | MATHEMATICS, APPLIED | FORMULAS

Journal Article

2.
Full Text
On the sum of the Lah numbers and zeros of the Kummer confluent hypergeometric function

Acta Universitatis Sapientiae, Mathematica, ISSN 1844-6094, 08/2018, Volume 10, Issue 1, pp. 125 - 133

In the paper, the authors find the sum of the Lah numbers and make sure that the Kummer confluent hypergeometric function (n+ 1; 2; z) has only n− 1real and...

Lah number | zero | Kummer confluent hypergeometric function | sum | Secondary 05A10, 05A19, 11B65, 11B99, 30C15, 33C15, 65B10 | Primary 11B83 | 65B10 | 33C15 | Secondary 05A10 | 11B65 | 05A19 | 11B99 | 30C15

Lah number | zero | Kummer confluent hypergeometric function | sum | Secondary 05A10, 05A19, 11B65, 11B99, 30C15, 33C15, 65B10 | Primary 11B83 | 65B10 | 33C15 | Secondary 05A10 | 11B65 | 05A19 | 11B99 | 30C15

Journal Article

Experimental Mathematics, ISSN 1058-6458, 10/2018, Volume 27, Issue 4, pp. 437 - 443

Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums are...

sums of powers of integers | Bernoulli numbers | Faulhaber's theorem | 11B65 | 11B68 | MATHEMATICS

sums of powers of integers | Bernoulli numbers | Faulhaber's theorem | 11B65 | 11B68 | MATHEMATICS

Journal Article

Journal of mathematical analysis and applications, ISSN 0022-247X, 2019, Volume 475, Issue 2, pp. 1636 - 1646

There are many instances known when the Fourier coefficients of modular forms are congruent to partial sums of hypergeometric series. In our previous work,...

Cyclotomic polynomial | Basic hypergeometric function | Ramanujan | (Super)congruence | q-Analogue | Q-ANALOG | 1/PI | MATHEMATICS | MATHEMATICS, APPLIED | RAMANUJAN-TYPE FORMULAS | SUPERCONGRUENCE | CONGRUENCES

Cyclotomic polynomial | Basic hypergeometric function | Ramanujan | (Super)congruence | q-Analogue | Q-ANALOG | 1/PI | MATHEMATICS | MATHEMATICS, APPLIED | RAMANUJAN-TYPE FORMULAS | SUPERCONGRUENCE | CONGRUENCES

Journal Article

Resultate der Mathematik, ISSN 1420-9012, 2019, Volume 75, Issue 1, pp. 1 - 15

We provide several new q-congruences for truncated basic hypergeometric series with the base being an even power of q. Our results mainly concern congruences...

Primary 33D15 | Andrews’ transformation | supercongruences | q -congruences | cyclotomic polynomial | 11B65 | Mathematics, general | Mathematics | Basic hypergeometric series | Secondary 11A07 | q-congruences

Primary 33D15 | Andrews’ transformation | supercongruences | q -congruences | cyclotomic polynomial | 11B65 | Mathematics, general | Mathematics | Basic hypergeometric series | Secondary 11A07 | q-congruences

Journal Article

6.
Full Text
A Kronecker‐type identity and the representations of a number as a sum of three squares

Bulletin of the London Mathematical Society, ISSN 0024-6093, 10/2017, Volume 49, Issue 5, pp. 770 - 783

By considering a limiting case of a Kronecker‐type identity, we obtain an identity found by both Andrews and Crandall. We then use the Andrews–Crandall...

11B65 (primary) | 11E25 | 05A15 (secondary) | FORMS | MATHEMATICS | YZ PLUS ZX | Mathematics - Number Theory

11B65 (primary) | 11E25 | 05A15 (secondary) | FORMS | MATHEMATICS | YZ PLUS ZX | Mathematics - Number Theory

Journal Article

Journal of the Mathematical Society of Japan, ISSN 0025-5645, 2014, Volume 66, Issue 3, pp. 927 - 949

Let v(d)((s)) denote the set of coefficient vectors of contractive polynomials of degree d with 2s non-real zeros. We prove that v(d)((s)) can be computed by a...

Polynomials with bounded roots | Selberg integral | MATHEMATICS | polynomials with bounded roots

Polynomials with bounded roots | Selberg integral | MATHEMATICS | polynomials with bounded roots

Journal Article

The Ramanujan journal, ISSN 1572-9303, 2018, Volume 49, Issue 3, pp. 531 - 544

Motivated by Zudilin’s work, we give q-analogues of the Ramanujan-type supercongruences (E.2) and (F.2) of Van Hamme. Our proof utilizes the q-WZ method and...

Cyclotomic polynomials | Ramanujan | 05A30 | Functions of a Complex Variable | Secondary 05A10 | Supercongruences | Field Theory and Polynomials | Primary 11B65 | Mathematics | q -Analogues | q -WZ pair | Fourier Analysis | Number Theory | Combinatorics | q-WZ pair | q-Analogues | MATHEMATICS | CONGRUENCES | Employee motivation

Cyclotomic polynomials | Ramanujan | 05A30 | Functions of a Complex Variable | Secondary 05A10 | Supercongruences | Field Theory and Polynomials | Primary 11B65 | Mathematics | q -Analogues | q -WZ pair | Fourier Analysis | Number Theory | Combinatorics | q-WZ pair | q-Analogues | MATHEMATICS | CONGRUENCES | Employee motivation

Journal Article

European Journal of Combinatorics, ISSN 0195-6698, 11/2016, Volume 58, pp. 247 - 266

Rook theory has been investigated by many people since its introduction by Kaplansky and Riordan in 1946. Goldman, Joichi, and White in 1975 showed that the...

MATHEMATICS | NUMBERS | FROBENIUS | Mathematics - Combinatorics

MATHEMATICS | NUMBERS | FROBENIUS | Mathematics - Combinatorics

Journal Article

Mathematica Slovaca, ISSN 0139-9918, 06/2018, Volume 68, Issue 3, pp. 501 - 512

Recently Marques and Trojovsky [On some new identities for the Fibonomial coefficients, Math. Slovaca 64 (2014), 809–818] presented interesting two sum...

Fibonomial coefficients | Gaussian | binomial coefficients | sum identities | Primary 11B65, 05A10, 11B37 | Gaussian q-binomial coefficients

Fibonomial coefficients | Gaussian | binomial coefficients | sum identities | Primary 11B65, 05A10, 11B37 | Gaussian q-binomial coefficients

Journal Article

Mathematika, ISSN 0025-5793, 01/2015, Volume 61, Issue 1, pp. 63 - 71

In this paper we present a new family of identities for multiple harmonic sums which generalize a recent result of Hessami Pilehrood et al [Trans. Amer. Math....

11B65 | 11M32 (primary) | MATHEMATICS | MATHEMATICS, APPLIED | MOTIVES | Mathematics - Number Theory

11B65 | 11M32 (primary) | MATHEMATICS | MATHEMATICS, APPLIED | MOTIVES | Mathematics - Number Theory

Journal Article

Mathematische Annalen, ISSN 0025-5831, 2/2017, Volume 367, Issue 1, pp. 373 - 395

Denote by p(n) the number of partitions of n and by N(a, M; n) the number of partitions of n with rank congruent to a modulo M. We find and prove a general...

11B65 | Mathematics, general | 11F27 | Mathematics | 11P84 | 11F11 | FORMS | MATHEMATICS | SYMMETRIES | SERIES | OVERPARTITIONS | PARTITIONS | MOCK THETA-FUNCTIONS | Polyimides | Mathematics - Number Theory

11B65 | Mathematics, general | 11F27 | Mathematics | 11P84 | 11F11 | FORMS | MATHEMATICS | SYMMETRIES | SERIES | OVERPARTITIONS | PARTITIONS | MOCK THETA-FUNCTIONS | Polyimides | Mathematics - Number Theory

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 04/2017, Volume 448, Issue 2, pp. 1061 - 1078

In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic...

Congruence | Pell number | Central binomial coefficient | LEGENDRE POLYNOMIALS | MATHEMATICS | MATHEMATICS, APPLIED | RODRIGUEZ-VILLEGAS | ANALOGS | HYPERGEOMETRIC-SERIES | CONGRUENCES | CONJECTURE

Congruence | Pell number | Central binomial coefficient | LEGENDRE POLYNOMIALS | MATHEMATICS | MATHEMATICS, APPLIED | RODRIGUEZ-VILLEGAS | ANALOGS | HYPERGEOMETRIC-SERIES | CONGRUENCES | CONJECTURE

Journal Article

Journal of difference equations and applications, ISSN 1563-5120, 2018, Volume 24, Issue 9, pp. 1473 - 1487

We prove two conjectures on sums of products of Catalan triangle numbers, which were originally conjectured by Miana et al. [Discrete Math. 340 (2017),...

cyclotomic polynomial | Zeilberger's algorithm | Catalan numbers | Catalan triangle | q-Chu-Vandermonde summation | 05A30 | 05A10 | 11B65 | MATHEMATICS, APPLIED | IDENTITIES | BINOMIAL COEFFICIENTS | POWER | MOMENTS | SUMS

cyclotomic polynomial | Zeilberger's algorithm | Catalan numbers | Catalan triangle | q-Chu-Vandermonde summation | 05A30 | 05A10 | 11B65 | MATHEMATICS, APPLIED | IDENTITIES | BINOMIAL COEFFICIENTS | POWER | MOMENTS | SUMS

Journal Article

Compositio mathematica, ISSN 0010-437X, 02/2018, Volume 154, Issue 2, pp. 249 - 274

We provide lower bounds for $p$ -adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb...

congruences | Apéry numbers | p-Lucas property | constant terms of powers of Laurent polynomials | MATHEMATICS | CONGRUENCE PROPERTIES | 11B65 | COEFFICIENTS | 05A10 (secondary) | 11B50 (primary) | Apery numbers | POWERS | Mathematics | Number Theory

congruences | Apéry numbers | p-Lucas property | constant terms of powers of Laurent polynomials | MATHEMATICS | CONGRUENCE PROPERTIES | 11B65 | COEFFICIENTS | 05A10 (secondary) | 11B50 (primary) | Apery numbers | POWERS | Mathematics | Number Theory

Journal Article

Journal of inequalities and applications, ISSN 1029-242X, 2018, Volume 2018, Issue 1, pp. 1 - 14

Recently, Mortenson (Proc. Edinb. Math. Soc. 4:1–13, 2015) explored the bilateral series in terms of Appell–Lerch sums for the universal mock theta function...

11F03 | Appell–Lerch sums | 11F27 | Bilateral series | Analysis | 11B65 | Mathematics, general | Mathematics | Applications of Mathematics | Mock theta functions | MATHEMATICS | MATHEMATICS, APPLIED | Appell-Lerch sums | Sums | Research

11F03 | Appell–Lerch sums | 11F27 | Bilateral series | Analysis | 11B65 | Mathematics, general | Mathematics | Applications of Mathematics | Mock theta functions | MATHEMATICS | MATHEMATICS, APPLIED | Appell-Lerch sums | Sums | Research

Journal Article

Duke Mathematical Journal, ISSN 0012-7094, 2001, Volume 108, Issue 3, pp. 395 - 419

As usual, define Dedekind's eta-function eta (z) by the infinite product [GRAPHICS] In a recent paper D. Zagier proved that (note: empty products equal 1...

MATHEMATICS | 11B65 | 11F20

MATHEMATICS | 11B65 | 11F20

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2018, Volume 2018, Issue 1, pp. 1 - 10

It is known that the concept of ratio monotonicity is closely related to log-convexity and log-concavity. In this paper, by exploring the log-behavior...

log-concavity | interlacing method | Analysis | 05A10 | 05A20 | 11B65 | Mathematics, general | Mathematics | log-convexity | Applications of Mathematics | ratio monotonicity | 11B37 | MATHEMATICS | MATHEMATICS, APPLIED | COMBINATORIAL SEQUENCES | Research

log-concavity | interlacing method | Analysis | 05A10 | 05A20 | 11B65 | Mathematics, general | Mathematics | log-convexity | Applications of Mathematics | ratio monotonicity | 11B37 | MATHEMATICS | MATHEMATICS, APPLIED | COMBINATORIAL SEQUENCES | Research

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2019, Volume 2019, Issue 1, pp. 1 - 11

In this paper, the authors introduce a new subclass of meromorphic q-starlike functions which are associated with the Janowski functions. A characterization of...

30C45 | 05A30 | Janowski functions | Meromorphically starlike functions | 11B65 | Mathematics | 47B38 | Analysis | Mathematics, general | Distortion theorems | Partial sums | Radius of q -starlikeness | Applications of Mathematics | q -derivative (or q -difference) | q-derivative (or q-difference) | Radius of q-starlikeness | MATHEMATICS | MATHEMATICS, APPLIED | Series expansion | Thermal expansion | Analytic functions | Mathematical analysis

30C45 | 05A30 | Janowski functions | Meromorphically starlike functions | 11B65 | Mathematics | 47B38 | Analysis | Mathematics, general | Distortion theorems | Partial sums | Radius of q -starlikeness | Applications of Mathematics | q -derivative (or q -difference) | q-derivative (or q-difference) | Radius of q-starlikeness | MATHEMATICS | MATHEMATICS, APPLIED | Series expansion | Thermal expansion | Analytic functions | Mathematical analysis

Journal Article

Arabian journal of mathematics, ISSN 2193-5343, 06/2018, Volume 7, Issue 2, pp. 101 - 112

We present three new sets of weighted partial sums of the Gaussian q-binomial coefficients. To prove the claimed results, we will use q-analysis, Rothe’s...

Binomial coefficients | Sums | 05A30 | 11B65

Binomial coefficients | Sums | 05A30 | 11B65

Journal Article

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