<abstract xmlns="http://www.w3.org/1999/xhtml">The purpose of this paper is to investigate identities with Jordan *-derivations in semiprime *-rings. Let ℛ be a 2-torsion free semiprime *-ring. In this paper it has been shown that, if <italic>ℛ</italic> admits an additive mapping <italic>D</italic> : <italic>ℛ→ℛ</italic>satisfying either <italic>D</italic>(<italic>xyx</italic>) = <italic>D</italic>(<italic>xy</italic>)<italic>x*</italic>+ <italic>xyD</italic>(<italic>x</italic>) for all <italic>x,y</italic> ∈ <italic>ℛ</italic>, or <italic>D</italic>(<italic>xyx</italic>) = <italic>D</italic>(<italic>x</italic>)<italic>y*x*</italic>+ <italic>xD</italic>(<italic>yx</italic>) for all pairs <italic>x, y</italic> ∈ <italic>ℛ</italic>, then <italic>D</italic> is a <italic>*</italic>-derivation. Moreover this result makes it possible to prove that if <italic>ℛ</italic> satis es 2<italic>D</italic>(<italic>xn</italic>) = <italic>D</italic>(<italic>xn−</italic>1)<italic>x*</italic> + <italic>xn−</italic>1<italic>D</italic>(<italic>x</italic>) + <italic>D</italic>(<italic>x</italic>)(<italic>x*</italic>)<italic>n−</italic>1 + <italic>xD</italic>(<italic>xn−</italic>1) for all <italic>x</italic> ∈ <italic>ℛ</italic> and some xed integer <italic>n ≥</italic> 2, then <italic>D</italic> is a Jordan <italic>*</italic>-derivation under some torsion restrictions. Finally, we apply these purely ring theoretic results to standard operator algebras <italic>𝒜</italic>(<italic>ℋ</italic>). In particular, we prove that if <italic>ℋ</italic> be a real or complex Hilbert space, with <italic>dim</italic>(<italic>ℋ</italic>) <italic>></italic> 1, admitting a linear mapping <italic>D</italic> : <italic>𝒜</italic>(<italic>ℋ</italic>) <italic>→ ℬ</italic>(<italic>ℋ</italic>) (where <italic>ℬ</italic>(<italic>ℋ</italic>) stands for the bounded linear operators) such that<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_puma-2015-0024_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mi>D</mi><mrow><mo>(</mo><mrow><msup><mi>A</mi><mi>n</mi></msup></mrow><mo>)</mo></mrow><mo>=</mo><mi>D</mi><mrow><mo>(</mo><mrow><msup><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><mo>)</mo></mrow><msup><mi>A</mi><mo>*</mo></msup><mo>+</mo><msup><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>D</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>+</mo><mi>D</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>A</mi><mo>*</mo></msup></mrow><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>A</mi><mi>D</mi><mrow><mo>(</mo><mrow><msup><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow></math><tex-math>$$2D\left( {A^n } \right) = D\left( {A^{n - 1} } \right)A^* + A^{n - 1} D\left( A \right) + D\left( A \right)\left( {A^* } \right)^{n - 1} + AD\left( {A^{n - 1} } \right)$$</tex-math></alternatives></disp-formula>for all <italic>A</italic>∈<italic>𝒜</italic>(<italic>ℋ</italic>). Then <italic>D</italic> is of the form <italic>D</italic>(<italic>A</italic>) = <italic>AB−BA*</italic> for all <italic>A</italic>∈<italic>𝒜</italic>(<italic>ℋ</italic>) and some fixed <italic>B</italic> ∈ <italic>ℬ</italic>(<italic>ℋ</italic>), which means that <italic>D</italic> is Jordan <italic>*</italic>-derivation.</abstract>

On certain functional equations related to Jordan <italic>*</italic>-derivations in semiprime <italic>*</italic>-rings and standard operator algebras

<abstract xmlns="http://www.w3.org/1999/xhtml">We investigate an equivalence relation on permutations based on the pure descent statistic. Generating functions are given for the number of equivalence classes for the set of all permutations, and the sets of permutations avoiding exactly one pattern of length three. As a byproduct, we exhibit a permutation set in one-to-one correspondence with forests of ordered binary trees, which provides a new combinatorial class enumerated by the single-source directed animals on the square lattice. Furthermore, bivariate generating functions for these sets are given according to various statistics.</abstract>

Forests and pattern-avoiding permutations modulo pure descents

<abstract xmlns="http://www.w3.org/1999/xhtml">Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence amenable to the kernel method.</abstract>

Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns

<abstract xmlns="http://www.w3.org/1999/xhtml">This paper is one of a series whose goal is to enumerate the avoiders, in the sense of classical pattern avoidance, for each triple of 4-letter patterns. There are 317 symmetry classes of triples of 4-letter patterns, avoiders of 267 of which have already been enumerated. Here we enumerate avoiders for all small Wilf classes that have a representative triple containing the pattern 1342, giving 40 new enumerations and leaving only 13 classes still to be enumerated. In all but one case, we obtain an explicit algebraic generating function that is rational or of degree 2. The remaining one is shown to be algebraic of degree 3. We use standard methods, usually involving detailed consideration of the left to right maxima, and sometimes the initial letters, to obtain an algebraic or functional equation for the generating function.</abstract>

Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns

<abstract xmlns="http://www.w3.org/1999/xhtml">In 1997, Richard André-Jeannin obtained a symmetric identity involving the reciprocal of the Horadam numbers <italic>Wn</italic>, defined by a three-term recurrence <italic>Wn</italic>+2 = <italic>P Wn</italic>+1<italic>− QWn</italic> with constant coefficients. In this paper, we extend this identity to sequences <italic>{an}n</italic>∈ℕ satisfying a three-term recurrence <italic>an</italic>+2 = <italic>pn</italic>+1<italic>an</italic>+1 + <italic>qn</italic>+1<italic>an</italic> with arbitrary coefficients. Then, we specialize such an identity to several <italic>q</italic>-polynomials of combinatorial interest, such as the <italic>q</italic>-Fibonacci, <italic>q</italic>-Lucas, <italic>q</italic>-Pell, <italic>q</italic>-Jacobsthal, <italic>q</italic>-Chebyshev and <italic>q</italic>-Morgan-Voyce polynomials.</abstract>

A generalization of André-Jeannin’s symmetric identity

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Pure Mathematics and Applications

Volume 27 (2018): Issue 1 (July 2018)

On certain functional equations related to Jordan -derivations in semiprime -rings and standard operator algebras

Forests and pattern-avoiding permutations modulo pure descents

Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns

Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns

A generalization of André-Jeannin’s symmetric identity

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Pure Mathematics and Applications

Volume 27 (2018): Issue 1 (July 2018)

On certain functional equations related to Jordan *-derivations in semiprime *-rings and standard operator algebras

Forests and pattern-avoiding permutations modulo pure descents

Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns

Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns

A generalization of André-Jeannin’s symmetric identity

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On certain functional equations related to Jordan -derivations in semiprime -rings and standard operator algebras