Summation Formulae for Binomial Moments
By combining the telescoping method with an algebraic relation, four classes of binomial moments are examined. Several explicit summation formulae are established.
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arXiv subjects
Wenchang Chu
Publications and source records attributed to Wenchang Chu.
Convolution Identities of Stirling Numbers
By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial identities are established as applications, that contain some well-known convolution formulae on Stirling numbers as special cases.
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Integrals of Hyperbolic Tangent Function
By means of the contour integration method, we evaluate, in closed form, a class of definite integrals involving hyperbolic tangent function.
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$q$-Analogues of $π$-Related Formulae from Jackson's $_8ϕ_7$-Series via Inversion Approach
By making use of the multiplicate form of the extended Carlitz inverse series relations, we establish two general `dual' theorems of Jackson's summation formula for well--poised $_8ϕ_7$-series. Their duplicate forms under the partition pattern $n=\lfloor{\frac{n}2}\rfloor+\lfloor{\frac{n+1}2}\rfloor$ are explored and yield numerous $q$-series identities whose limiting cases as $q\to1$ result in classical $π$-related Ramanujan--like series of convergence rate ``$\frac1{16}$" including one for $1/π^2$ discovered by Guillera (2003). The triplicate dual formulae under the partition pattern $n=\lfloor{\frac{n}3}\rfloor+\lfloor{\frac{n+1}3}\rfloor+\lfloor{\frac{n+2}3}\rfloor$ are examined via the ``reverse bisection method", which leads us to twenty new $q$-series identities together with their classical counterparts of convergence rate ``$\frac{-1}{27}$" when $q\to1$.
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Triplicate Dual Series of Dougall--Dixon Theorem
Applying the triplicate form of the extended Gould--Hsu inverse series relations to Dougall's summation theorem for the well--poised $_7F_6$-series, we establish, from the dual series, several interesting Ramanujan--like infinite series expressions for $π^2$ and $π^{\pm1}$ with convergence rate "$-\frac{1}{27}$".
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$π$-Formulae from Dual Series of the Dougall Theorem
By means of the extended Gould-Hsu inverse series relations, we find that the dual relation of Dougall's summation theorem for the well--poised $_7F_6$-series can be utilized to construct numerous interesting Ramanujan--like infinite series expressions for $π^{\pm1}$ and $π^{\pm2}$, including an elegant formula of $π^{-2}$ due to Guillera.
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Alternating Convolutions of Catalan Numbers
A new class of alternating convolutions concerning binomial coefficients and Catalan numbers are evaluated in closed forms.
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$q$-Analogues of $π$-Series by Applying Carlitz Inversions to $q$-Pfaff-Saalsch{ü}tz Theorem
By applying multiplicate forms of the Carlitz inverse series relations to the $q$-Pfaff-Saalsch{ü}tz summation theorem, we establish twenty five nonterminating $q$-series identities with several of them serving as $q$-analogues of infinite series expressions for $π$ and $1/π$, including some typical ones discovered by Ramanujan (1914) and Guillera.
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Partial Sums of Two Quartic q-Series
The partial sums of two quartic basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several summation and transformation formulae are consequently established.
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Finite Form of the Quintuple Product Identity
The celebrated quintuple product identity follows surprisingly from an almost-trivial algebraic identity, which is the limiting case of the terminating q-Dixon formula.
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Partial-Fraction Decompositions and Harmonic Number Identities
By means of partial fraction method, we investigate the decomposition of rational functions. Several striking identities on harmonic numbers and generalized Apery numbers will be established, including the binomial-harmonic number identity associated with Beukers' conjecture on Apery numbers.
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Hypergeometric Series and Harmonic Number Identities
The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers including those discovered recently by Paule and Schneider (2003).
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