Published Papers and Preprints

• New harmonic number identities with applications .
  arxiv:912.2497 , (13/12/2009).
  We determine the explicit formulas for the sum of products of homogeneous multiple harmonic sums $\sum_{k=1}^n \prod_{j=1}^r H_k(\{1\}^{\lambda_j})$ when $\sum_{j=1}^r \lambda_j\leq 5$. We apply these identities to the study of two congruences modulo a power of a prime.

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• An elementary proof of a Rodriguez-Villegas supercongruence .
  arXiv:0911.4261 , (22/11/2009).
  We give a short proof of the following known congruence: for every odd prime $p$ $$\sum_{k=0}^{p-1}{2k\choose k}^2 16^{-k}\equiv (-1)^{{p-1\over 2}}\pmod{p^2}.$$ Moreover, we provide some new results connected with the above congruence.

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• Congruences of multiple sums involving invariant sequences under binomial transform .
  arXiv:0911.1074v1 , (05/11/2009) submitted for publication.
  We will prove several congruences modulo a power of a prime such as $$\sum_{0\max(n+1,3)$.

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• Congruences of alternating multiple harmonic sums (with Jianqiang Zhao).
  arXiv:0909.0670v1 , (03/09/2009) submitted for publication.
  In this sequel to arXiv:0905.3327, we continue to study the congruence properties of the alternating version of multiple harmonic sums. As contrast to the study of multiple harmonic sums where Bernoulli numbers and Bernoulli polynomials play the key roles, in the alternating setting the Euler numbers and the Euler polynomials are also essential.

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• Congruences involving the reciprocals of central binomial coefficients.
  arXiv:0906.5150v1 , (28/06/2009) submitted for publication.
  We present several congruences modulo a power of prime $p$ concerning sums of the following type $\sum_{k=1}^{p-1}{m^k\over k^r}{2k\choose k}^{-1}$ which reveal some interesting connections with the analogous infinite series.

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• q-Analogs of some congruences involving Catalan numbers.
  arXiv:0905.3816v1 , (23/05/2009) submitted for publication.
  We provide some variations on the Greene-Krammer's identity which involve q-Catalan numbers. Our method reveals a curious analogy between these new identities and some congruences modulo a prime.

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• Congruences involving alternating multiple harmonic sum.
  arXiv:0905.3327v2 , (21/05/2009) submitted for publication.
  We show that for any prime prime $p\not=2$ $ $\sum_{k=1}^{p-1} {(-1)^k\over k}{-{1\over 2} \choose k} \equiv -\sum_{k=1}^{(p-1)/2}{1\over k} \pmod{p^3}$$ by expressing the l.h.s. as a combination of alternating multiple harmonic sums.

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• New congruences for central binomial coefficients (with Zhi-Wei Sun).
  arXiv:0805.0563 , (26/03/2009) submitted for publication.
  Let $p$ be a prime, and let $d\in\{0,...,p^a\}$ with $a\in\Z^+$. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ where $m$ is an integer not divisible by $p$. For example, we show that if $p\not=2,5$ then $$\sum_{k=1}^{p-1}(-1)^k\frac{\binom{2k}k}k=-5\frac{F_{p-(\frac p5)}}p (mod p),$$ where $F_n$ denotes the $n$th Fibonacci number. We also prove that if $p>3$ then $$\sum_{k=1}^{p-1}\frac{\binom{2k}k}k={8/9} p^2B_{p-3} (mod p^3),$$ where $B_n$ is the $n$th Bernoulli number.

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• Edge Cover Time for Regular Graphs.
  Journal of Integer Sequences, 11, 8.4.4 (2008).
  Consider the following stochastic process on a graph: initially all vertices are uncovered and at each step cover the two vertices of a random edge. What is the expected number of steps required to cover all vertices of the graph? In this note we show that the mean cover time for a regular graph of N vertices is asymptotically (N log N)/2. Moreover, we compute the exact mean cover time for some regular graphs via generating functions.

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• The operad Lie is free (with Paolo Salvatore).
  arXiv:0802.3010 , (21/02/2008) accepted for publication in Journal of Pure and Applied Algebra.
  We show that the operad Lie is free as a non-symmetric operad. Then we study the generating series counting the operadic generators, finding a recursive formula for its coefficients, and showing that the asymptotic density of the operadic generators is 1/e (see the corresponding new sequence A134988 in Neil Sloane's Online Encyclopedia of Integer Sequences.

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• Congruences involving Catalan numbers (with Zhi-Wei Sun).
  arXiv:0709.1665 , (24/09/2007) submitted for publication.
  In this paper we establish some new congruences involving Catalan numbers as well as central binomial coefficients. Let p>3 be a prime. We show that sum_{k=0}^{p-1}C_{pn+k}/C_n=1-3(n+1)((p-1)/3) (mod p^2) for every n=0,1,2,..., where C_m is the Catalan number binom(2m,m)/(m+1), and (*/3) is the Legendre symbol. We also determine sum_{k=0}^{p^a-1}binom(2k,k+d) and sum_{k=0}^{p^a-1}k*binom(2k,k+d) modulo p^2 for all a=1,2,3,... and d=0,1,...,p.

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• The dinner table problem: the rectangular case.
  INTEGERS, A11, 6 (2006).
  $n$ people are seated randomly at a rectangular table with $\lfloor n/2\rfloor$ and $\lceil n/2\rceil$ seats along the two opposite sides for two dinners. What's the probability that neighbors at the first dinner are no more neighbors at the second one? We give an explicit formula and we show that its asymptotic behavior as $n$ goes to infinity is $e^{-2}(1+4/n)$ (it is known that it is $e^{-2}(1-4/n)$ for a round table). See the corresponding new sequence A110128 in Neil Sloane's Online Encyclopedia of Integer Sequences.

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• Rigidity of holomorphic generators and one-parameter semigroups (with Mark Elin, Marina Levenshtein and David Shoikhet).
  Dynamic Systems and Applications, vol.16 (2007),no.2, 251-266
  In this paper we establish a rigidity property of holomorphic generators by using their local behavior at a boundary point $\tau$ of the open unit disk $\Delta$. Namely, if $f\in\mathrm{Hol}(\Delta,\mathbb{C})$ is the generator of a one-parameter continuous semigroup $\{F_{t}\}_{t\geq0}$, we state that the equality $f(z)=o\left(|z-\tau|^{3}\right)$ when $z\rightarrow\tau$ in each non-tangential approach region at $\tau$ implies that $f$ vanishes identically on $\Delta$. Note, that if $F$ is a self-mapping of $\Delta$ then $f=I-F$ is a generator, so our result extends the boundary version of the Schwarz Lemma obtained by D. Burns and S. Krantz.

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• Congruences for Sums of Binomial Coefficients (with Zhi-Wei Sun).
  Journal of Number Theory, 126(2007), no.2, 287-296
  We find an explicit integer nu_m(q) such that [n+ nu_m(q) ,r]_m=[n,r ]_m (mod q) where [n,r]_m=sum_{k=r mod m} binomial(n,k). This is a further extension of a congruence of Glaisher.

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• A Common Fixed Point Theorem for Commuting Expanding Maps on Nilmanifolds.
  Electronic Journal of Differential Equations, Conf. 12 (2005), 181-188.
  A self-map f of a compact connected manifold M is expanding if it locally expands distances with respect to some metric. We consider the case when M is a nilmanifold and we discuss a new common fixed point theorem for two expanding maps which commute.

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• A New Domino Tiling Sequence.
  Journal of Integer Sequences, 7, 2.3 (2004).
  In this short note, we prove that the sequence A061646 in Neil Sloane's Online Encyclopedia of Integer Sequences is connected with the number of domino tilings of a holey square.

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• Commuting Holomorphic Self-Maps of the Unit Disc.
  Ergodic Theory and Dynamical Systems, 24, 945-953 (2004).
  Let f,g be two commuting holomorphic self-maps of the unit disc. If f and g are fixed point free and they agree at the common Wolff point up to third order of derivatives then f and g are identically equal. Using the linear model of Baker and Pommerenke we improve a previous result for the parabolic case achieving a more ''natural'' smoothness conditions.

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• The Lindelof principle and angular derivatives in convex domains of finite type (with Marco Abate).
  Journal of the Australian Mathematical Society, 73, 1-30 (2002).
  We describe a generalization of the classical Julia-Wolff-Caratheodory theorem to a large class of bounded convex domains of finite type, including convex circular domains and convex domains with real analytic boundary. The main tools used in the proofs are several explicit estimates on the boundary behaviour of Kobayashi distance and metric, and a new Lindelof principle.

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• Identity Principles for Commuting Holomorphic Self-Maps of the Unit Disc (with Filippo Bracci and Fabio Vlacci).
  Journal of Mathematical Analysis and Applications, 270, 451-473 (2002).
  Let f,g be two commuting holomorphic self-maps of the unit disc. If f and g agree at the common Wolff point up to a certain order of derivatives (no more than 3 if the Wolff point is on the unit circle), then f and g are identically equal.

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• Rigidity at the Boundary for Holomorphic Self-Maps of the Unit Disc (with Fabio Vlacci).
  Complex Variables, 45, 151-165 (2001).
  We prove a rigidity theorem which generalizes a result due to Burns and Krantz for holomorphic self-maps in the unit disc of the complex plane. The authors found that some conditions on the (boundary) Schwarzian derivative of a holomorphic self-map at specific points of the boundary of the disc may be sufficient to conclude that the map is a completely determined rational map.

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• The Julia-Wolff-Caratheodory theorem(s) (with Marco Abate).
  Kim, Kang-Tae (ed.) et al., Complex geometric analysis in Pohang. POSTECH-BSRI SNU-GARC international conference on several complex variables, Pohang, Korea, June 23-27, 1997. Providence, RI: American Mathematical Society. Contemp. Math., 222, 161-172 (1999).
  It is described a general framework allowing generalization of the Julia-Wolff-Caratheodory theorem to several classes of bounded domain in C^n. As an example, the authors discuss the case of bounded convex circular domains.

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• Sets of Periods for Expanding Maps on Flat Manifolds.
  Monatshefte für Mathematik 128, 151-157 (1999).
  It is proven that the sets of periods for expanding maps on n-dimensional flat manifolds are uniformly cofinite, i.e. there is a positive integer m_0, which depends only on n,such that for any integer m>=m_0, for any n-dimensional flat manifold M and for any expanding map F on M, there exists a periodic point of F whose least period is exactly m.

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• Common fixed points of commuting holomorphic maps of the polydisc which are expanding on the torus.
  Advances in Mathematics, 138, No.1, 92-104 (1998).
  The following result is proved. Let F and G be two holomorphic maps of open polydisc in C^n which are continuous on the closure, map the torus in itself and are expanding on the torus. If F and G commute on the torus then they have a unique fixed point in the open polydisc.

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• Centralizers of polynomials.
  Rendiconti Ist. Mat. Univ. Trieste 28, No.1-2, 63-69 (1996).
  The centralizer of a nonlinear polynomial P in C [z] is the set of nonlinear polynomials Q such that P(Q(z))=Q(P(z)). The centralizer is trivial if it consists only of the iterates of P. The author shows that there is an open dense subset S of the nonlinear polynomials whose elements have trivial centralizers. In fact if U is the set of polynomials of degree d>1 which have (d+1) different fixed points (including infinity) with different eigenvalues, then S=U\{T_2,-T_2\}, T_2(z) =2z^2-1.

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• On fixed points of C^1 extensions of expanding maps in the unit disc.
  Atti dell'Accademia Nazionale dei Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rendiconti Lincei, Mat. Appl., 5, No.4, 303-308 (1994).
  Using a result due to M. Shub, a theorem about the existence of fixed points inside the unit disc for C^1 extensions of expanding maps defined on the boundary is established.

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• On fixed points of holomorphic maps of simply connected proper domains in C.
  Atti dell'Accademia Nazionale dei Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rendiconti Lincei, Mat. Appl., 5, No.2, 197-202 (1994).
  A sufficient condition of the existence of fixed point of one-dimensional holomorphic map is given. If f has at least three zeros or a zero with multiplicity >=3 in {z: |z|<1/2}, then f satisfies the sufficient condition.

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• La matematica del domino.
  A divulgative paper about domino tiling problems that I've written in memory of Professor Franco Conti.
  It has been published in the book Ricordando Franco Conti , Scuola Normale Superiore, Pisa, 2005.

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