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Volumn 15, Issue 1, 2011, Pages 285-305

Some new families of generalized Euler and Genocchi polynomial

Author keywords

Apostol bernoulli polynomials; Apostol euler polynomials; Apostol genocchi polynomials; Bernoulli polynomials; Euler polynomials; Gauss summation theorem; Gaussian hypergeometric function; Genocchi polynomials; Hurwitz lerch zeta function; Leibniz rule; Pfaff kummer transformation; Stirling numbers of the second kind; Taylor maclaurin series expansion

Indexed keywords


EID: 79251611867     PISSN: 10275487     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article
Times cited : (57)

References (48)
  • 1
    • 0003851729 scopus 로고
    • Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Tenth Printing
    • M. Abramowitz and I. A. Stegun (Editors), National Bureau of Standards, Washington D.C
    • M. Abramowitz and I. A. Stegun (Editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Tenth Printing, Applied Mathematics Series 55, National Bureau of Standards, Washington D.C., 1972.
    • (1972) Applied Mathematics Series , vol.55
  • 2
    • 84972529289 scopus 로고
    • On the Lerch Zeta function
    • T. M. Apostol, On the Lerch Zeta function, Pacific J. Math., 1 (1951), 161-167.
    • (1951) Pacific J. Math , vol.1 , pp. 161-167
    • Apostol, T.M.1
  • 3
    • 42649116806 scopus 로고    scopus 로고
    • Some q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n and the multiple Hurwitz zeta function
    • J. Choi, P. J. Anderson and H. M. Srivastava, Some q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n and the multiple Hurwitz zeta function, Appl. Math. Comput., 199 (2008), 723-737.
    • (2008) Appl. Math. Comput , vol.199 , pp. 723-737
    • Choi, J.1    Anderson, P.J.2    Srivastava, H.M.3
  • 4
    • 69249216343 scopus 로고    scopus 로고
    • Carlitz's q-Bernoulli and q-Euler numbers and polynomials and a class of q-Hurwitz zeta functions
    • J. Choi, P. J. Anderson and H. M. Srivastava, Carlitz's q-Bernoulli and q-Euler numbers and polynomials and a class of q-Hurwitz zeta functions, Appl. Math. Comput., 215 (2009), 1185-1208.
    • (2009) Appl. Math. Comput , vol.215 , pp. 1185-1208
    • Choi, J.1    Anderson, P.J.2    Srivastava, H.M.3
  • 6
    • 33749589040 scopus 로고    scopus 로고
    • Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch Zeta functions
    • M. Garg, K. Jain and H. M. Srivastava, Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch Zeta functions, Integral Transforms Spec. Funct., 17 (2006), 803-815.
    • (2006) Integral Transforms Spec. Funct , vol.17 , pp. 803-815
    • Garg, M.1    Jain, K.2    Srivastava, H.M.3
  • 7
    • 3042834009 scopus 로고    scopus 로고
    • On the generalized Riemann Zeta functions and the generalized Lambert transform
    • S. P. Goyal and R. K. Laddha, On the generalized Riemann Zeta functions and the generalized Lambert transform, Gan. ita Sandesh, 11 (1997), 99-108.
    • (1997) Gan. Ita Sandesh , vol.11 , pp. 99-108
    • Goyal, S.P.1    Laddha, R.K.2
  • 8
    • 0013306271 scopus 로고    scopus 로고
    • Generalization of Bernoulli polynomials, Internat
    • B.-N. Guo and F. Qi, Generalization of Bernoulli polynomials, Internat. J. Math. Ed. Sci. Tech., 33 (2002), 428-431.
    • (2002) J. Math. Ed. Sci. Tech , vol.33 , pp. 428-431
    • Guo, B.-N.1    Qi, F.2
  • 10
    • 79251609138 scopus 로고
    • Negative order Genocchi polynomials
    • A. F. Horadam, Negative order Genocchi polynomials, Fibonacci Quart., 30 (1992), 21-34.
    • (1992) Fibonacci Quart , vol.30 , pp. 21-34
    • Horadam, A.F.1
  • 11
    • 27144448168 scopus 로고
    • Generation of Genocchi polynomials of first order by recurrence relations
    • A. F. Horadam, Generation of Genocchi polynomials of first order by recurrence relations, Fibonacci Quart., 30 (1992), 239-243.
    • (1992) Fibonacci Quart , vol.30 , pp. 239-243
    • Horadam, A.F.1
  • 12
    • 70349205736 scopus 로고    scopus 로고
    • Interpolation functions of q-extensions of Apostol's type Euler polynomials
    • 2009, Article ID
    • K.-W. Hwang, Y.-H. Kim and T. Kim, Interpolation functions of q-extensions of Apostol's type Euler polynomials, J. Inequal. Appl., 2009 (2009), Article ID 451217, 1-12.
    • (2009) J. Inequal. Appl , pp. 1-12
    • Hwang, K.-W.1    Kim, Y.-H.2    Kim, T.3
  • 13
    • 61449220268 scopus 로고    scopus 로고
    • On the distribution of the q-Euler polynomials and the q-Genocchi polynomials of higher order
    • 2008, Article ID
    • L.-C. Jang and T. Kim, On the distribution of the q-Euler polynomials and the q-Genocchi polynomials of higher order, J. Inequal. Appl., 2008 (2008), Article ID 723615, 1-9.
    • (2008) J. Inequal. Appl , pp. 1-9
    • Jang, L.-C.1    Kim, T.2
  • 14
    • 33751172538 scopus 로고    scopus 로고
    • On the q-extension of Euler numbers and Genocchi numbers
    • T. Kim, On the q-extension of Euler numbers and Genocchi numbers, J. Math. Anal. Appl., 326 (2007), 1458-1465.
    • (2007) J. Math. Anal. Appl , vol.326 , pp. 1458-1465
    • Kim, T.1
  • 15
    • 57849089562 scopus 로고    scopus 로고
    • On the multiple q-Genocchi and Euler numbers, Russian
    • T. Kim, On the multiple q-Genocchi and Euler numbers, Russian J. Math. Phys., 15 (2008), 481-486.
    • (2008) J. Math. Phys , vol.15 , pp. 481-486
    • Kim, T.1
  • 16
    • 58149522391 scopus 로고    scopus 로고
    • On the Q-extension of Apostol-Euler Numbers and Polynomials
    • Article ID
    • Y.-H. Kim, W.-J. Kim and L.-C. Jang, On the q-extension of Apostol-Euler numbers and polynomials, type=journal>Abstr. Appl. Anal., 2008 (2008), Article ID 296159, 1-10.
    • (2008) Abstr. Appl. Anal , pp. 1-10
    • Kim, Y.-H.1    Kim, W.-J.2    Jang, L.-C.3
  • 17
    • 77950944052 scopus 로고    scopus 로고
    • A further symmetric relation on the analogue of the Apostol-Bernoulli and the analogue of the Apostol-Genocchi polynomials
    • B. Kurt, A further symmetric relation on the analogue of the Apostol-Bernoulli and the analogue of the Apostol-Genocchi polynomials, Appl. Math. Sci. (Ruse), 3 (2009), 2757-2764.
    • (2009) Appl. Math. Sci. (Ruse) , vol.3 , pp. 2757-2764
    • Kurt, B.1
  • 18
    • 77952012578 scopus 로고    scopus 로고
    • The multiplication formulae for the Genocchi polynomials
    • V. Kurt, The multiplication formulae for the Genocchi polynomials, Proc. Jangjeon Math. Soc., 13 (2010), 89-96.
    • (2010) Proc. Jangjeon. Math. Soc , vol.13 , pp. 89-96
    • Kurt, V.1
  • 19
    • 77958572572 scopus 로고    scopus 로고
    • A new approach to q-Genocchi numbers and polynomials
    • V. Kurt and M. Cenkci, A new approach to q-Genocchi numbers and polynomials, Bull. Korean Math. Soc. 47 (2010), 575-583.
    • (2010) Bull. Korean Math. Soc , vol.47 , pp. 575-583
    • Kurt, V.1    Cenkci, M.2
  • 20
    • 3042752343 scopus 로고    scopus 로고
    • Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations
    • S.-D. Lin and H. M. Srivastava, Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput., 154 (2004), 725-733.
    • (2004) Appl. Math. Comput , vol.154 , pp. 725-733
    • Lin, S.-D.1    Srivastava, H.M.2
  • 21
    • 33749664753 scopus 로고    scopus 로고
    • Some expansion formulas for a class of generalized Hurwitz-Lerch Zeta functions
    • S.-D. Lin, H. M. Srivastava and P.-Y. Wang, Some expansion formulas for a class of generalized Hurwitz-Lerch Zeta functions, Integral Transforms Spec. Funct., 17 (2006), 817-828.
    • (2006) Integral Transforms Spec. Funct , vol.17 , pp. 817-828
    • Lin, S.-D.1    Srivastava, H.M.2    Wang, P.-Y.3
  • 22
    • 67349223315 scopus 로고    scopus 로고
    • Some identities on the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums
    • H.-M. Liu and W.-P. Wang, Some identities on the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums, Discrete Math., 309 (2009), 3346-3363.
    • (2009) Discrete Math , vol.309 , pp. 3346-3363
    • Liu, H.-M.1    Wang, W.-P.2
  • 23
    • 19044397438 scopus 로고    scopus 로고
    • On the Apostol-Bernoulli polynomials, Central European
    • Q.-M. Luo, On the Apostol-Bernoulli polynomials, Central European J. Math., 2 (2004), 509-515.
    • (2004) J. Math , vol.2 , pp. 509-515
    • Luo, Q.-M.1
  • 24
    • 33747419087 scopus 로고    scopus 로고
    • Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions
    • Q.-M. Luo, Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J. Math., 10 (2006), 917-925.
    • (2006) Taiwanese J. Math , vol.10 , pp. 917-925
    • Luo, Q.-M.1
  • 25
    • 70349863230 scopus 로고    scopus 로고
    • Fourier Expansions and Integral Representations for the Apostol-Bernoulli and Apostol-Euler Polynomials
    • Q.-M. Luo, Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials, Math. Comput., 78 (2009), 2193-2208.
    • (2009) Math. Comput , vol.78 , pp. 2193-2208
    • Luo, Q.-M.1
  • 26
    • 68949114125 scopus 로고    scopus 로고
    • The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order
    • Q.-M. Luo, The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order, Integral Transforms Spec. Funct., 20 (2009), 377-391.
    • (2009) Integral Transforms Spec. Funct , vol.20 , pp. 377-391
    • Luo, Q.-M.1
  • 27
    • 59249090149 scopus 로고    scopus 로고
    • Fourier expansions and integral representations for the Genocchi polynomials
    • Article 09.1.4
    • Q.-M. Luo, Fourier expansions and integral representations for the Genocchi polynomials, J. Integer Seq., 12 (2009), Article 09.1.4, 1-9.
    • (2009) J. Integer Seq , vol.12 , pp. 1-9
    • Luo, Q.-M.1
  • 28
    • 75749103911 scopus 로고    scopus 로고
    • Q-Extensions for the Apostol-Genocchi polynomials
    • Q.-M. Luo, q-Extensions for the Apostol-Genocchi polynomials, Gen. Math., 17 (2009), 113-125.
    • (2009) Gen. Math , vol.17 , pp. 113-125
    • Luo, Q.-M.1
  • 29
    • 78650926144 scopus 로고    scopus 로고
    • Some formulas for Apostol-Euler polynomials associated with Hurwitz zeta function at rational arguments
    • Q.-M. Luo, Some formulas for Apostol-Euler polynomials associated with Hurwitz zeta function at rational arguments, Appl. Anal. Discrete Math., 3 (2009), 336-346.
    • (2009) Appl. Anal. Discrete Math , vol.3 , pp. 336-346
    • Luo, Q.-M.1
  • 30
    • 79551634909 scopus 로고    scopus 로고
    • Extension for the Genocchi polynomials and its Fourier expansions and integral representations
    • to appear
    • Q.-M. Luo, Extension for the Genocchi polynomials and its Fourier expansions and integral representations, Osaka J. Math., (to appear).
    • Osaka J. Math
    • Luo, Q.M.1
  • 31
    • 13844251489 scopus 로고    scopus 로고
    • Generalizations of Bernoulli numbers and polynomials, Internat
    • Q.-M. Luo, B.-N. Guo, F. Qi and L. Debnath, Generalizations of Bernoulli numbers and polynomials, Internat. J. Math. Math. Sci., 2003 (59) (2003), 3769-3776.
    • (2003) J. Math. Math. Sci , vol.2003 , Issue.59 , pp. 3769-3776
    • Luo, Q.-M.1    Guo, B.-N.2    Qi, F.3    Debnath, L.4
  • 32
    • 17844378174 scopus 로고    scopus 로고
    • Generalizations of Euler numbers and polynomials, Internat
    • Q.-M. Luo, F. Qi and L. Debnath, Generalizations of Euler numbers and polynomials, Internat. J. Math. Math. Sci., 2003 (61) (2003), 3893-3901.
    • (2003) J. Math. Math. Sci , vol.2003 , Issue.61 , pp. 3893-3901
    • Luo, Q.-M.1    Qi, F.2    Debnath, L.3
  • 33
    • 19044388957 scopus 로고    scopus 로고
    • Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials
    • Q.-M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308 (2005), 290-302.
    • (2005) J. Math. Anal. Appl , vol.308 , pp. 290-302
    • Luo, Q.-M.1    Srivastava, H.M.2
  • 34
    • 33745719864 scopus 로고    scopus 로고
    • Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials
    • Q.-M. Luo and H. M. Srivastava, Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl., 51 (2006), 631-642.
    • (2006) Comput. Math. Appl , vol.51 , pp. 631-642
    • Luo, Q.-M.1    Srivastava, H.M.2
  • 35
    • 4243995184 scopus 로고
    • Springer-Verlag, Berlin, 1924; Reprinted by Chelsea Publishing Company, Bronx, New York
    • N. E. N̈orlund, Vorlesungen ̈ uber Differentzenrechnung, Springer-Verlag, Berlin, 1924; Reprinted by Chelsea Publishing Company, Bronx, New York, 1954.
    • (1954) Vorlesungen ̈ Uber Differentzenrechnung
    • N̈orlund, N.E.1
  • 36
    • 78049294653 scopus 로고    scopus 로고
    • A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials
    • H. Ozden, Y. Simsek and H. M. Srivastava, A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl., 60 (2010), 2779-2787.
    • (2010) Comput. Math. Appl , vol.60 , pp. 2779-2787
    • Ozden, H.1    Simsek, Y.2    Srivastava, H.M.3
  • 37
    • 75449101143 scopus 로고    scopus 로고
    • Pad́e approximation and Apostol-Bernoulli and Apostol-Euler polynomials
    • M. Pŕevost, Pad́e approximation and Apostol-Bernoulli and Apostol-Euler polynomials, J. Comput. Appl. Math., 233 (2010), 3005-3017.
    • (2010) J. Comput. Appl. Math , vol.233 , pp. 3005-3017
    • Pŕevost, M.1
  • 38
    • 0003863948 scopus 로고    scopus 로고
    • Kluwer Academic Publishers, Dordrecht, Boston and London
    • J. Śandor and B. Crstici, Handbook of Number Theory, Vol. II, Kluwer Academic Publishers, Dordrecht, Boston and London, 2004.
    • (2004) Handbook of Number Theory , vol.2
    • Śandor, J.1    Crstici, B.2
  • 39
    • 23044520680 scopus 로고    scopus 로고
    • Some formulas for the Bernoulli and Euler polynomials at rational arguments
    • H. M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc., 129 (2000), 77-84.
    • (2000) Math. Proc. Cambridge Philos. Soc , vol.129 , pp. 77-84
    • Srivastava, H.M.1
  • 41
    • 77953460296 scopus 로고    scopus 로고
    • A new generalization of the Bernoulli and related polynomials, Russian
    • H. M. Srivastava, M. Garg and S. Choudhary, A new generalization of the Bernoulli and related polynomials, Russian J. Math. Phys., 17 (2010), 251-261.
    • (2010) J. Math. Phys , vol.17 , pp. 251-261
    • Srivastava, H.M.1    Garg, M.2    Choudhary, S.3
  • 42
    • 0004162942 scopus 로고
    • Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto
    • H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.
    • (1984) A Treatise on Generating Functions
    • Srivastava, H.M.1    Manocha, H.L.2
  • 43
    • 1842790483 scopus 로고    scopus 로고
    • H. M. Srivastava and A ́. Pinte ́r, Remarks on some relationships between the ernoulli and Euler polynomials, Appl. Math. Lett., 17 (2004), 375-380.
    • (2004) Math. Lett , vol.17 , pp. 375-380
    • Srivastava, H.M.1    Pintér, A.2
  • 44
    • 0000381058 scopus 로고
    • An explicit formula for the generalized ernoulli polynomials
    • H. M. Srivastava and P. G. Todorov, An explicit formula for the generalized ernoulli polynomials, J. Math. Anal. Appl., 130 (1988), 509-513.
    • (1988) J. Math. Anal. Appl , vol.130 , pp. 509-513
    • Srivastava, H.M.1    Todorov, P.G.2
  • 45
    • 38849106160 scopus 로고    scopus 로고
    • Some results on the Apostol-Bernoulli and Apostol-Euler polynomials
    • W.-P. Wang, C.-Z. Jia and T.-M. Wang, Some results on the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl., 55 (2008), 1322-1332.
    • (2008) Comput. Math. Appl , vol.55 , pp. 1322-1332
    • Wang, W.-P.1    Jia, C.-Z.2    Wang, T.-M.3
  • 46
    • 77951017339 scopus 로고    scopus 로고
    • Some results on power sums and Apostol-type polynomials
    • W.-P. Wang and W.-W. Wang, Some results on power sums and Apostol-type polynomials, Integral Transforms Spec. Funct., 21 (2010), 307-318.
    • (2010) Integral Transforms Spec. Funct , vol.21 , pp. 307-318
    • Wang, W.-P.1    Wang, W.-W.2
  • 47
    • 0032222377 scopus 로고    scopus 로고
    • Some identities involving generalized Genocchi polynomials and generalized Fibonacci-Lucas sequences
    • Z.-Z. Zhang and J.-Y. Jin, Some identities involving generalized Genocchi polynomials and generalized Fibonacci-Lucas sequences, Fibonacci Quart., 36 (1998), 329-334.
    • (1998) Fibonacci Quart , vol.36 , pp. 329-334
    • Zhang, Z.-Z.1    Jin, J.-Y.2
  • 48
    • 55649096851 scopus 로고    scopus 로고
    • Several identities for the generalized Apostol-Bernoulli polynomials
    • Z.-Z. Zhang and H.-Q. Yang, Several identities for the generalized Apostol-Bernoulli polynomials, Comput. Math. Appl., 56 (2008), 2993-2999.
    • (2008) Comput. Math. Appl , vol.56 , pp. 2993-2999
    • Zhang, Z.-Z.1    Yang, H.-Q.2


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