Some Generalizations and Basic (or q-) Extensions of the Bernoulli, Euler

In the vast literature in Analytic Number Theory, one can find systematic and extensive investigations not only of the classical Bernoulli, Euler and Genocchi polynomials and their corresponding numbers, but also of their many generalizations and basic (or q-) extensions. Our main object in this presentation is to introduce and investigate some of the principal generalizations and unifications of each of these polynomials by means of suitable generating functions. We present several interesting properties of these general polynomial systems including some explicit series representations in terms of the Hurwitz (or generalized) zeta function and the familiar Gauss hypergeometric function. By introducing an analogue of the Stirling numbers of the second kind, that is, the so-called -Stirling numbers of the second kind, we derive several properties and formulas and consider some of their interesting applications to the family of the Apostol type polynomials. We also give a brief expository and historial account of the various basic (or q-) extensions of the classical Bernoulli polynomials and numbers, the classical Euler polynomials and numbers, the classical Genocchi polynomials and numbers, and also of their such generalizations as (for example) the above-mentioned families of the Apostol type polynomials and numbers. Relevant connections of the definitions and results presented here with those in earlier as well as forthcoming investigations will be indicated.

[1]  Hari M. Srivastava,et al.  Carlitz's q-Bernoulli and q-Euler numbers and polynomials and a class of generalized q-Hurwitz zeta functions , 2009, Appl. Math. Comput..

[2]  Hari M. Srivastava,et al.  A new generalization of the Bernoulli and related polynomials , 2010 .

[3]  T. Apostol.  Introduction to analytic number theory , 1976 .

[4]  Qiu-Ming Luo,et al.  The multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order , 2009 .

[5]  Hari M. Srivastava,et al.  Some New Families of Generalized Euler and Genocchi Polynomials , 2011 .

[6]  Taekyun Kim,et al.  On the multiple q-Genocchi and Euler numbers , 2008, 0801.0978.

[7]  Waleed A. Al-Salam,et al.  q‐Bernoulli numbers and polynomials , 1958 .

[8]  Hari M. Srivastava,et al.  A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials , 2010, Comput. Math. Appl..

[9]  Hari M. Srivastava,et al.  Some formulas for the Bernoulli and Euler polynomials at rational arguments , 2000, Mathematical Proceedings of the Cambridge Philosophical Society.

[10]  Yilmaz Simsek,et al.  A new extension of q-Euler numbers and polynomials related to their interpolation functions , 2008, Appl. Math. Lett..

[11]  H. M. Srivastava,et al.  Some expansion formulas for a class of generalized Hurwitz–Lerch Zeta functions , 2006 .

[12]  Jacek Klinowski,et al.  New formulae for the Bernoulli and Euler polynomials at rational arguments , 1995 .

[13]  Hari M. Srivastava,et al.  Integral and computational representations of the extended Hurwitz–Lerch zeta function , 2011 .

[14]  Marc Prévost,et al.  Padé approximation and Apostol-Bernoulli and Apostol-Euler polynomials , 2010, J. Comput. Appl. Math..

[15]  E. Verspohl.  The pharmacy and pharmacotherapy of asthma. Ed. by P. F. D'Arcy and J. C. McElnay. Ellis Horwood Series in Pharmaceutical Technology. Halsted Press: A division of John Wiley & Sons, New York/Chichester/Brisbane/Toronto. 258 S., zahlr. Abb., geb. £ 49.95 , 1990 .

[16]  Lee-Chae Jang,et al.  On the Distribution of the -Euler Polynomials and the -Genocchi Polynomials of Higher Order , 2008 .

[17]  Takashi Agoh,et al.  On Bernoulli and Euler numbers , 1988 .

[18]  Taekyun Kim.  q-Generalized Euler numbers and polynomials , 2006 .

[19]  H. Srivastava,et al.  Some extensions of the Fermi-Dirac and Bose-Einstein functions with applications to the family of the zeta and related functions , 2011 .

[20]  Hari M. Srivastava,et al.  Series Associated with the Zeta and Related Functions , 2001 .

[21]  Hari M. Srivastava,et al.  An explicit formula for the generalized Bernoulli polynomials , 1988 .

[22]  Taekyun Kim,et al.  Interpolation Functions of -Extensions of Apostol's Type Euler Polynomials , 2009 .

[23]  Khristo N. Boyadzhiev.  Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials , 2007 .

[24]  T. Kim,et al.  q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients , 2008 .

[25]  Y. Simsek,et al.  $q$-Bernoulli Numbers and Polynomials Associated with Multiple $q$-Zeta Functions and Basic $L$-series , 2005, math/0502019.

[26]  Lee-Chae Jang,et al.  A note on $q$-Euler and Genocchi numbers , 2001 .

[27]  Feng Qi (祁锋),et al.  Generalization of Bernoulli polynomials , 2002 .

[28]  Weiping Wang,et al.  Some identities on the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums , 2009, Discret. Math..

[29]  Taekyun Kim,et al.  ON THE ANALOGS OF BERNOULLI AND EULER NUMBERS, RELATED IDENTITIES AND ZETA AND L-FUNCTIONS , 2008 .

[30]  Hari M. Srivastava,et al.  Remarks on some relationships between the Bernoulli and Euler polynomials , 2004, Appl. Math. Lett..

[31]  L. Carlitz.  Expansions of $q$-Bernoulli numbers , 1958 .

[32]  Hari M. Srivastava,et al.  Some q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n, and the multiple Hurwitz zeta function , 2008, Appl. Math. Comput..

[33]  P. W. Karlsson,et al.  Multiple Gaussian hypergeometric series , 1985 .

[34]  Yilmaz Simsek,et al.  q-Analogue of twisted l-series and q-twisted Euler numbers , 2005 .

[35]  Veli Kurt,et al.  A NEW APPROACH TO q-GENOCCHI NUMBERS AND POLYNOMIALS , 2010 .

[36]  Zhizheng Zhang,et al.  Several identities for the generalized Apostol-Bernoulli polynomials , 2008, Comput. Math. Appl..

[37]  Veli Kurt.  A Further Symmetric Relation on the Analogue of the Apostol-Bernoulli and the Analogue of the Apostol-Genocchi Polynomials , 2009 .

[38]  H. Srivastava,et al.  q-Extensions of Some Relationships Between the Bernoulli and Euler Polynomials , 2011 .

[39]  Leonard Carlitz,et al.  $q$-Bernoulli numbers and polynomials , 1948 .

[40]  Lokenath Debnath,et al.  GENERALIZATIONS OF BERNOULLI NUMBERS AND POLYNOMIALS , 2003 .

[41]  K. S. Kölbig,et al.  Errata: Milton Abramowitz and Irene A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, No. 55, U.S. Government Printing Office, Washington, D.C., 1994, and all known reprints , 1972 .

[42]  Qiu-Ming Luo,et al.  Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials , 2009, Math. Comput..

[43]  Weiping Wang,et al.  Some results on power sums and Apostol-type polynomials , 2010 .

[44]  A. W. Kemp,et al.  A treatise on generating functions , 1984 .

[45]  Lee-Chae Jang,et al.  On the q-Extension of Apostol-Euler Numbers and Polynomials , 2008 .

[46]  Tom M. Apostol,et al.  On the Lerch zeta function. , 1951 .

[47]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[48]  József Sándor,et al.  Handbook of Number Theory I , 1995 .

[49]  Taekyun Kim,et al.  Some formulae for the q-Bernoulli and Euler polynomials of higher order , 2002 .

[50]  Taekyun Kim,et al.  Barnes-type multiple q-zeta functions and q-Euler polynomials , 2009, 0912.5119.

[52]  Hari M. Srivastava,et al.  Some generalizations of the Apostol–Bernoulli and Apostol–Euler polynomials , 2005 .

[53]  Qiu-Ming Luo,et al.  Fourier Expansions and Integral Representations for Genocchi Polynomials , 2009 .

[54]  Taekyun Kim,et al.  The modified q-Euler numbers and polynomials , 2008 .

[55]  Tianming Wang,et al.  Some results on the Apostol-Bernoulli and Apostol-Euler polynomials , 2008, Comput. Math. Appl..

[56]  Qiu-Ming Luo,et al.  Extensions of the Genocchi polynomials and their Fourier expansions and integral representations , 2011 .

[57]  Zhi-Wei Sun,et al.  SOME IDENTITIES FOR BERNOULLI , 2004 .

[58]  Leonard Carlitz,et al.  $q$-Bernoulli and Eulerian numbers , 1954 .

[59]  Hari M. Srivastava,et al.  Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials , 2006, Comput. Math. Appl..

[60]  Hari M. Srivastava,et al.  Some relationships between the generalized Apostol–Bernoulli polynomials and Hurwitz–Lerch Zeta functions , 2006 .

[61]  Hari M. Srivastava,et al.  Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind , 2011, Appl. Math. Comput..

[62]  L. Comtet,et al.  Advanced Combinatorics: The Art of Finite and Infinite Expansions , 1974 .

[63]  Qiu-Ming Luo,et al.  q-Extensions for the Apostol-Genocchi Polynomials , 2009 .

[64]  Taekyun Kim,et al.  On the q-extension of Euler and Genocchi numbers , 2007 .

[65]  Hari M. Srivastava,et al.  The Multiple Hurwitz Zeta Function and the Multiple Hurwitz-Euler Eta Function , 2011 .

[66]  Cenkci Mehmet,et al.  q-EXTENSIONS OF GENOCCHI NUMBERS , 2006 .

[67]  Qiu-Ming Luo,et al.  APOSTOL-EULER POLYNOMIALS OF HIGHER ORDER AND GAUSSIAN HYPERGEOMETRIC FUNCTIONS , 2006 .

[68]  Shy-Der Lin,et al.  Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations , 2004, Appl. Math. Comput..

[69]  Yilmaz Simsek,et al.  Twisted (h,q)-Bernoulli numbers and polynomials related to twisted (h,q)-zeta function and L-function☆ , 2006 .

[70]  Yilmaz Simsek,et al.  On (i, q) Bernoulli and Euler numbers , 2008, Appl. Math. Lett..

[71]  Qiu-Ming Luo.  On the apostol-bernoulli polynomials , 2004 .