In 1969 Mocanu [1] introduced and studied a new class of analytic functions consisting of
<mml:math alttext="$\alpha $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#945;</mml:mi></mml:math>-convex functions. Many mathematicians have studied and shown the properties of this class. Now we
will define new classes like that Mocanu class and then investigate and give some results. The class of <mml:math alttext="$\alpha $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#945;</mml:mi></mml:math>-convex functions of order <mml:math alttext="$\beta $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#946;</mml:mi></mml:math> partially includes Mocanu's class.

Seiichi  Fukui

On <svg style="vertical-align:-0.1638pt" height="7.9499998" version="1.1" viewBox="0 0 9.7749996 7.9499998" width="9.7749996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D6FC" d="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204&#xA;q-14 98 -31 149.5t-50 51.5q-49 0 -94 -70t-45 -164q0 -55 15.5 -86t40.5 -31q70 0 164 150z" /></g></svg>-convex functions of order <svg style="vertical-align:-3.56265pt" height="16.6625" version="1.1" viewBox="0 0 10 16.6625" width="10" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D6FD" d="M558 587q0 -32 -14 -61t-40 -53.5t-48.5 -41t-54.5 -36.5q144 -51 144 -174q0 -55 -43.5 -108t-104.5 -87q-77 -42 -131 -42q-31 0 -54 20t-31 47l11 18q48 -29 108 -29q79 0 119.5 43t40.5 109t-44.5 107.5t-119.5 50.5l22 47q34 1 65 21q96 61 96 157q0 42 -24 67.5&#xA;t-62 25.5q-24 0 -43.5 -9t-35 -29.5t-27 -44t-22.5 -63t-19.5 -75.5t-18.5 -91q-57 -294 -68 -380q-26 -190 -35 -200q-26 -31 -97 -37l-4 26q19 9 48 170l77 413q23 121 52.5 187.5t83.5 114.5q70 62 148 62q51 0 88.5 -34t37.5 -91z" /></g></svg> 

In this paper we study those <mml:math alttext="$n$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-color partitions of Agarwal and Andrews, 1987, in which
each pair of parts has weighted difference equal to <mml:math alttext="$ - 2$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:math> Results obtained in this paper for these
partitions include several combinatorial identities, recurrence relations, generating functions, relationships
with the divisor function and computer produced tables. By using these partitions an explicit expression
for the sum of the divisors of odd integers is given. It is shown how these partitions arise in the study of
conjugate and self-conjugate <mml:math alttext="$n$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-color partitions. A combinatorial identity for self-conjugate <mml:math alttext="$n$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-color
partitions is also obtained. We conclude by posing several open problems in the last section.

A. K. Agarwal

R. Balasubrananian

<i>n</i>-Color partitions with weighted differences equal to minus two

For a sequence-to-sequence transformation <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>, let <mml:math alttext="$R_m Ax = \sum\nolimits_{n \geqslant m} {\left| {\left( {Ax} \right)_n } \right|} $ " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mi>A</mml:mi><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mo>&#8721;</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8805;</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:math> and <mml:math alttext="$\mu _m Ax = \sup _{n \geqslant m} \left| {\left( {Ax} \right)_n } \right|$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#956;</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mi>A</mml:mi><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mo>sup</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8805;</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:math>. The purpose of this paper is to study the relationship between the
asymptotic equivalence of two sequences <mml:math alttext="$\left( {\lim _n {{x_n } \mathord{\left/  {\vphantom {{x_n } {y_n  = 1}}} \right.  \kern-\nulldelimiterspace} {y_n  = 1}}} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mo>lim</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> and the variations of asymptotic
equivalence based on the ratios <mml:math alttext="${{R_m Ax} \mathord{\left/  {\vphantom {{R_m Ax} {R_m Ay}}} \right.  \kern-\nulldelimiterspace} {R_m Ay}}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mi>A</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mi>A</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mrow></mml:math> and <mml:math alttext="${{\mu _m Ax} \mathord{\left/  {\vphantom {{\mu _m Ax} {\mu _m Ay}}} \right.  \kern-\nulldelimiterspace} {\mu _m Ay}}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mi>&#956;</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mi>A</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>&#956;</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mi>A</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mrow></mml:math>.

Jinlu  Li

Asymptotic equivalence of sequences and summability

Let <mml:math alttext="$G$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math> be a simple graph of order at least three. We show that <mml:math alttext="$G$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math> is Hamiltonian-connected
if and only if its strong closure is Hamiltonian-connected. We also give an efficient
algorithm to compute the strong closure of <mml:math alttext="$G$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math>.

Pak-Ken  Wong

Hamiltonian-connected graphs and their strong closures

King[3] introduced and examined the concepts of almost A-summable sequence,
almost conservative matrix and almost regular matrix By following King, in this paper we introduce
and examine the concepts of <mml:math alttext="$\theta $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#952;</mml:mi></mml:math>-almost A-summable sequence, <mml:math alttext="$\theta $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#952;</mml:mi></mml:math>-almost conservative matrix and <mml:math alttext="$\theta $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#952;</mml:mi></mml:math>-almost regular matrix

Fatih  Nuray


<svg style="vertical-align:-0.1638pt" height="12.4375" version="1.1" viewBox="0 0 8.5874996 12.4375" width="8.5874996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D703" d="M475 507q0 -83 -20 -172t-56 -167.5t-93.5 -129t-125.5 -50.5q-157 0 -157 227q0 78 21.5 164t59 161t96.5 123.5t126 48.5q79 0 114 -58t35 -147zM391 522q0 155 -81 155q-62 0 -111 -82.5t-73 -200.5h253q12 81 12 128zM373 346h-255q-12 -91 -12 -150q0 -72 20 -123&#xA;t63 -51q34 0 64 28.5t52.5 77t39 103t28.5 115.5z" /></g></svg>-Almost summable sequences

Let <mml:math alttext="$N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> be a semiprime right near-ring, <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> a subset of <mml:math alttext="$N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> such that <mml:math alttext="$0 \in A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo>&#8712;</mml:mo><mml:mi>A</mml:mi></mml:math> and <mml:math alttext="$AN \subseteq A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi><mml:mi>N</mml:mi><mml:mo>&#10949;</mml:mo><mml:mi>A</mml:mi></mml:math>, and <mml:math alttext="$d$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math> a derivation of <mml:math alttext="$N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> The purpose of this paper is to prove that if d acts as a homomorphism on <mml:math alttext="$A$ " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> or as an anti-homomorphism on <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>, then <mml:math alttext="$d\left( A \right) = \left\{ 0 \right\}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>A</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mn>0</mml:mn><mml:mo>}</mml:mo></mml:mrow></mml:math>.

Nurcan  Arga&#231;

On prime and semiprime near-rings with derivations

The multiple Hermite series in <mml:math alttext="$R^n $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi mathvariant="bold">R</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math> are investigated by the Riesz summability
method of order <mml:math alttext="$\alpha  &#62; {{\left( {n - 1} \right)} \mathord{\left/  {\vphantom {{\left( {n - 1} \right)} 2}} \right.  \kern-\nulldelimiterspace} 2}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#945;</mml:mi><mml:mo>&#62;</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>. More precisely, localization theorems for some classes of functions
are proved and sharp sufficient conditions are given. Thus the classical Szeg&#246; results are extended
to the <mml:math alttext="$n$s" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-dimensional case. In particular, for these classes of functions the localization principle
and summability on the Lebesgue set are established.

G. E. Karadzhov

E. E. El-Adad

Localization and summability of multiple Hermite series

A method is outlined to express a Turán determinant of solutions of a three term
recurrence relation as a weighted sum of squares. This method is shown to imply the positivity of Turán determinants of symmetric Pollaczek polynomials, Lommel polynomials and <mml:math alttext="$q$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math>-Bessel functions.

Joaquin Bustoz

Mourad E. H. Ismail

Turán inequalities for symmetric orthogonal polynomials

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International Journal of Mathematics and Mathematical Sciences publishes research across all fields of mathematics and mathematical sciences, such as pure and applied mathematics, mathematical physics, probability and mathematical statistics.

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International Journal of Mathematics and Mathematical Sciences

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International Journal of Mathematics and Mathematical Sciences