• Kyungpook Mathematical Journal
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• v.49 no.3
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• pp.483-492
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• 2009

In this paper we give formulae for sums of products of two Horadam type generalized Fibonacci numbers with the same recurrence equation and with possibly different initial conditions. Analogous improved alternating sums are also studied as well as various derived sums when terms are multiplied either by binomial coefficients or by members of the sequence of natural numbers. These formulae are related to the recent work of Belbachir and Bencherif, $\v{C}$erin and $\v{C}$erin and Gianella.

• Speech Sciences
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• v.10 no.4
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• pp.7-21
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• 2003

Sums-of-Products models were built for segment duration prediction of spoken Korean. An experiment for the modelling was carried out to apply the results to Korean text-to-speech synthesis systems. 670 read sentences were analyzed. trained and tested for the construction of the duration models. Traditional sequential rule systems were extended to simple additive, multiplicative and additive-multiplicative models based on Sums-of-Products modelling. The parameters used in the modelling include the properties of the target segment and its neighbors and the target segment's position in the prosodic structure. Two optimisation strategies were used: the downhill simplex method and the simulated annealing method. The performance of the models was measured by the correlation coefficient and the root mean squared prediction error (RMSE) between actual and predicted duration in the test data. The best performance was obtained when the data was trained and tested by ' additive-multiplicative models. ' The correlation for the vowel duration prediction was 0.69 and the RMSE. 31.80 ms. while the correlation for the consonant duration prediction was 0.54 and the RMSE. 29.02 ms. The results were not good enough to be applied to the real-time text-to-speech systems. Further investigation of feature interactions is required for the better performance of the Sums-of-Products models.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.401-408
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• 2014

We establish a new approach for the sums of products of the Euler numbers by using the relation of values at non-positive integers of the representation of the multiple Hurwitz-Euler eta function in terms of the Hurwitz-Euler eta function.

• Journal of the Korean Institute of Intelligent Systems
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• v.11 no.2
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• pp.173-177
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• 2001

We introduce cascade products, wreath products, sums and joins of TL-finite state machines and investigate their algebraic structures. Also we study the relations with other products of TL-finite state machines.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1389-1413
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• 2013

In this paper, we consider several convolution sums, namely, $\mathcal{A}_i(m,n;N)$ ($i=1,2,3,4$), $\mathcal{B}_j(m,n;N)$ ($j=1,2,3$), and $\mathcal{C}_k(m,n;N)$ ($k=1,2,3,{\cdots},12$), and establish certain identities involving their finite products. Then we extend these types of product convolution identities to products involving Faulhaber sums. As an application, an identity involving the Weierstrass ${\wp}$-function, its derivative and certain linear combination of Eisenstein series is established.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.319-331
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• 2019

On the weighted Dirichlet space, by the Sobolev's embedding theorem, we characterize the compactness for operators which are finite sums of products of several Toeplitz operators. Moreover, the essential spectrum of Toeplitz operator is characterized.

• Journal of the Korean Mathematical Society
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• v.33 no.1
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• pp.57-68
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• 1996

In the classical analysis there are various theorems which permit us to interchange limits and infinite sums, limits and integrals, integrals and infinite sums, etc. The infinite products as well as the infinite series play an important role in different branches of mathematics.

• Communications of the Korean Mathematical Society
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• v.25 no.2
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• pp.215-224
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• 2010

Generalized nonnegative polynomials are defined as products of nonnegative polynomials raised to positive real powers. The generalized degree can be defined in a natural way. In this paper we extend quadrature sums involving pth powers of polynomials to those for generalized polynomials.

• Journal of the Korean Mathematical Society
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• v.57 no.4
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• pp.845-871
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• 2020

We consider generalized Dedekind sums in dimension n, defined as sum of products of values of periodic Bernoulli functions. For the generalized Dedekind sums, we associate a Laurent polynomial. Using this, we associate an exponential sum of a Laurent polynomial to the generalized Dedekind sums and show that this exponential sum has a nontrivial bound that is sufficient to fulfill the equidistribution criterion of Weyl and thus the fractional part of the generalized Dedekind sums are equidistributed in ℝ/ℤ.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.985-1001
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• 2023

Recently, Alzer and Choi [2] introduced and studied a set of the four linear Euler sums with parameters. These sums are parametric extensions of Flajolet and Salvy's four kinds of linear Euler sums [9]. In this paper, by using the method of residue computations, we will establish two explicit combined formulas involving two parametric linear Euler sums S⁺⁺_p,q (a, b) and S^+-_p,q (a, b) defined by Alzer and Choi, which can be expressed in terms of a linear combinations of products of trigonometric functions, digamma functions and Hurwitz zeta functions.