Note on <inline-formula> <graphic file="1029-242X-2008-371295-i1.gif"/></inline-formula>-Extensions of Euler Numbers and Polynomials of Higher Order
<p>Abstract</p> <p>In 2007, Ozden et al. constructed generating functions of higher-order twisted <inline-formula> <graphic file="1029-242X-2008-371295-i2.gif"/></inline-formula>-extension of Euler polynomials and numbers, by using <inline-formula>...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2008-01-01
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| Series: | Journal of Inequalities and Applications |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/2008/371295 |
| Summary: | <p>Abstract</p> <p>In 2007, Ozden et al. constructed generating functions of higher-order twisted <inline-formula> <graphic file="1029-242X-2008-371295-i2.gif"/></inline-formula>-extension of Euler polynomials and numbers, by using <inline-formula> <graphic file="1029-242X-2008-371295-i3.gif"/></inline-formula>-adic, <inline-formula> <graphic file="1029-242X-2008-371295-i4.gif"/></inline-formula>-deformed fermionic integral on <inline-formula> <graphic file="1029-242X-2008-371295-i5.gif"/></inline-formula>. By applying their generating functions, they derived the complete sums of products of the twisted <inline-formula> <graphic file="1029-242X-2008-371295-i6.gif"/></inline-formula>-extension of Euler polynomials and numbers. In this paper, we consider the new <inline-formula> <graphic file="1029-242X-2008-371295-i7.gif"/></inline-formula>-extension of Euler numbers and polynomials to be different which is treated by Ozden et al. From our <inline-formula> <graphic file="1029-242X-2008-371295-i8.gif"/></inline-formula>-Euler numbers and polynomials, we derive some interesting identities and we construct <inline-formula> <graphic file="1029-242X-2008-371295-i9.gif"/></inline-formula>-Euler zeta functions which interpolate the new <inline-formula> <graphic file="1029-242X-2008-371295-i10.gif"/></inline-formula>-Euler numbers and polynomials at a negative integer. Furthermore, we study Barnes-type <inline-formula> <graphic file="1029-242X-2008-371295-i11.gif"/></inline-formula>-Euler zeta functions. Finally, we will derive the new formula for "sums of products of <inline-formula> <graphic file="1029-242X-2008-371295-i12.gif"/></inline-formula>-Euler numbers and polynomials" by using fermionic <inline-formula> <graphic file="1029-242X-2008-371295-i13.gif"/></inline-formula>-adic, <inline-formula> <graphic file="1029-242X-2008-371295-i14.gif"/></inline-formula>-integral on <inline-formula> <graphic file="1029-242X-2008-371295-i15.gif"/></inline-formula>.</p> |
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| ISSN: | 1025-5834 1029-242X |