• Communications of the Korean Mathematical Society
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• v.16 no.2
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• pp.319-332
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• 2001

The main object of this paper is first to give tow contiguous analogues of a well-known hypergeometric summation formula for $_2$F$_1$(1/2). We then apply each of these analogues with a view to evaluating the sums of several classes of series in terms of Psi(or Digamma) and the Zeta functions. Relevant connections of the series identities presented here with those given elsewhere are also pointed out.

• The Pure and Applied Mathematics
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• v.8 no.2
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• pp.127-135
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• 2001

The main object of this paper is to present a transformation formula for a finite series involving $_3F_2$ and some identities associated with the binomial coefficients by making use of the theory of Legendre polynomials $P_{n}$(x) and some summation theorems for hypergeometric functions $_pF_q$. Some integral formulas are also considered.

• East Asian mathematical journal
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• v.34 no.5
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• pp.597-600
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• 2018

Bailey presented two Bailey presented two interesting identities involving $_0F_1$, which have been generalized by Choi and Rathie who used two hypergeometric summation formulas due to Qureshi et al. In this note, we aim to show how one can establish, in an elementary way, two generalized formulas involving $_0F_1$ which include the above-mentioned identities as special cases. interesting identities involving 0F1, which

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.329-342
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• 2016

Recently, Korobov polynomials have been received a lot of attention, which are discrete analogs of Bernoulli polynomials. In particular, these polynomials are used to derive some interpolation formulas of many variables and a discrete analog of the Euler summation formula. In this paper, we extend these family of polynomials to consider the Korobov polynomials of the fifth kind and of the sixth kind. We present several explicit formulas and recurrence relations for these polynomials. Also, we establish a connection between our polynomials and several known families of polynomials.

• Honam Mathematical Journal
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• v.32 no.1
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• pp.61-71
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• 2010

The aim of this paper is to extend a number of transformation formulas for the four $X_4$, $X_5$, $X_7$, and $X_8$ among twenty triple hypergeometric series $X_1$ to $X_{20}$ introduced earlier by Exton. The results are derived from the generalized Kummer's theorem and Dixon's theorem obtained earlier by Lavoie et al..

• Honam Mathematical Journal
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• v.31 no.2
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• pp.167-176
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• 2009

The aim of this paper is to obtain three interesting results for reducibility of Kamp$\'{e}$ de $\'{e}$riet function. The results are derived with the help of contiguous Gauss's second summation formulas obtained earlier by Lavoie et al. The results obtained by Bailey, Rathie and Nagar follow special cases of our main findings.

• Journal of the Korean Mathematical Society
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• v.56 no.1
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• pp.265-284
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• 2019

The main purpose of this paper is to investigate the q-Apostol type Frobenius-Euler numbers and polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive q-integers. By using infinite series representation for q-Apostol type Frobenius-Euler numbers and polynomials including their interpolation functions, we not only give some identities and relations for these numbers and polynomials, but also define generating functions for new numbers and polynomials. Further we give remarks and observations on generating functions for these new numbers and polynomials. By using these generating functions, we derive recurrence relations and finite sums related to these numbers and polynomials. Moreover, by applying higher-order derivative to these generating functions, we derive some new formulas including the Hurwitz-Lerch zeta function, the Apostol-Bernoulli numbers and the Apostol-Euler numbers. Finally, for an application of the generating functions, we derive a multiplication formula, which is very important property in the theories of normalized polynomials and Dedekind type sums.

• Honam Mathematical Journal
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• v.26 no.1
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• pp.119-132
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• 2004

In 1965, Bhatt and Pandey have obtained an analogue of the Whipple's theorem for double series by using Watson's theorem on the sum of a $_3F_2$. The aim of this paper is to derive twenty five results for double series closely related to the analogue of the Whipple's theorem for double series obtained by Bhatt and Pandey. The results are derived with the help of twenty five summation formulas closely related to the Watson's theorem on the sum of a $_3F_2$ obtained recently by Lavoie, Grondin, and Rathie.

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.1951-1967
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• 2017

We aim to introduce a further extension of a family of the extended Hurwitz-Lerch Zeta functions of two variables. We then systematically investigate several interesting properties of the extended function such as its integral representations which provide extensions of various earlier corresponding results of two and one variables, its summation formula, its Mellin-Barnes type contour integral representations, its computational representation and fractional derivative formulas. A multi-parameter extension of the extended Hurwitz-Lerch Zeta function of two variables is also introduced. Relevant connections of certain special cases of the main results presented here with some known identities are pointed out.

• Communications of the Korean Mathematical Society
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• v.24 no.2
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• pp.291-302
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• 2009

Contiguous relations for hypergeometric series contain an enormous amount of hidden information. Applications of contiguous relations range from the evaluation of hypergeometric series to the derivation of summation and transformation formulas for such series. In this paper, a general formula joining three Gauss functions of the form $_2F_1$[$a_1$, $a_2$; $a_3$; z] with arbitrary integer shifts is presented. Our analysis depends on using shifted operators attached to the three parameters $a_1$, $a_2$ and $a_3$. We also, discussed the existence condition of our formula.