• Communications of the Korean Mathematical Society
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• v.35 no.1
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• pp.21-41
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• 2020

In this paper, we establish an explicit formula for r-Dowling numbers in terms of r-Whitney Lah and r-Whitney numbers of the second kind. This is a generalization of the Qi formula for Bell numbers in terms of Lah and Stirling numbers of the second kind. Moreover, we define the q, r-Dowling numbers, q, r-Whitney Lah numbers and q, r-Whitney numbers of the first kind and obtain several fundamental properties of these numbers such as orthogonality and inverse relations, recurrence relations, and generating functions. Hence, we derive an analogous Qi formula for q, r-Dowling numbers expressed in terms of q, r-Whitney Lah numbers and q, r-Whitney numbers of the second kind.

• Journal of the Korean Mathematical Society
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• v.47 no.3
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• pp.645-657
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• 2010

In this paper, more generalized q-factorial coefficients are examined by a natural extension of the q-factorial on a sequence of any numbers. This immediately leads to the notions of the extended q-Stirling numbers of both kinds and the extended q-Lah numbers. All results described in this paper may be reduced to well-known results when we set q = 1 or use special sequences.

• Journal of the Korean Mathematical Society
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• v.49 no.3
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• pp.537-547
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• 2012

In this paper, we define a $p$, $q$-difference operator and obtain an explicit formula which is used to express the $p$, $q$-analogue of the unified generalization of Stirling numbers and its exponential generating function in terms of the $p$, $q$-difference operator. Explicit formulas for the non-central $q$-Stirling numbers of the second kind and non-central $q$-Lah numbers are derived using the new $q$-analogue of Newton's interpolation formula. Moreover, a $p$, $q$-analogue of Newton's interpolation formula is established.

• Clinical and Experimental Reproductive Medicine
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• v.41 no.2
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• pp.68-74
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• 2014

Objective: In search of an ideal method of assisted hatching (AH), we compared the effects of conventional micropipette-AH and laser-AH on the blastocyst formation rate (BFR) and blastocyst cell numbers. Methods: Four-to five-week-old ICR female mice were paired with male mice after superovulation using Pregnant mare's serum gonadotropin (PMSG) and hCG. The two-cell embryos were flushed from the oviducts of female mice. The retrieved two-cell embryos underwent one of five AH procedures: single mechanical assisted hatching (sMAH); cross mechanical assisted hatching (cMAH); single laser assisted hatching (sLAH); quarter laser assisted hatching (qLAH); and quarter laser zona thinning assisted hatching (qLZT-AH). After 72 hours incubation, double immunofluorescence staining was performed. Results: Following a 72 hours incubation, a higher hatching BFR was observed in the control, sMAH, cMAH, and sLAH groups, compared to those in the qLAH and qLZT-AH groups (p<0.05). The hatched BFR was significantly higher in the qLAH and qLZT-AH groups than in the others (p<0.05 for each group). The inner cell mass (ICM) was higher in the control and sMAH group (p<0.05). The trophectoderm cell number was higher in the cMAH and qLAH groups (p<0.05). Conclusion: Our results showed that the hatched BFR was higher in groups exposed the the qLAH and qLZT-AH methods compared to groups exposed to other AH methods. In the qLAH group, although the total cell number was significantly higher than in controls, the ICM ratio was significantly lower in than controls.