Research ArticleAnalytic Continuation of Euler Polynomialsand the Euler Zeta Function
C S Ryoo
Department of Mathematics Hannam University Daejeon 306-791 Republic of Korea
Correspondence should be addressed to C S Ryoo ryoocshnukr
Received 10 January 2014 Accepted 10 March 2014 Published 3 April 2014
Academic Editor Binggen Zhang
Copyright copy 2014 C S Ryoo This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We study that the Euler numbers 119864119899and Euler polynomials 119864
119899(119911) are analytically continued to 119864 (119904) and 119864(119904 119908) We investigate
the new concept of dynamics of the zeros of analytica continued polynomials Finally we observe an interesting phenomenon ofldquoscatteringrdquo of the zeros of 119864(119904 119908)
1 Introduction
Throughout this paper Z R and C will denote the ring ofintegers the field of real numbers and the complex numbersrespectively Recently many mathematicians have studieddifferent kinds of the Euler Bernoulli andGenocchi numbersand polynomials (see [1ndash19]) The computing environmentwould make more and more rapid progress and there hasbeen increasing interest in solving mathematical problemswith the aid of computers By using software mathematicianscan explore concepts much more easily than in the pastThe ability to create and manipulate figures on the computerscreen enables mathematicians to quickly visualize and pro-duce many problems examine properties of the figures lookfor patterns and make conjectures This capability is espe-cially exciting because these steps are essential formostmath-ematicians to truly understand even basic concept Numeri-cal experiments of Bernoulli polynomials Euler polynomialsand Genocchi polynomials have been the subject of extensivestudy in recent years and much progress has been made bothmathematically and computationally Using computer a real-istic study for Euler polynomials119864
119899(119909) is very interesting It is
the aim of this paper to observe an interesting phenomenonof ldquoscatteringrdquo of the zeros of the Euler polynomials 119864
119899(119909) in
complex plane First we introduce the Euler numbers andEuler polynomials As a well-known definition the Eulernumbers 119864
119899are defined by
2
119890119905 + 1=
infin
sum
119899=0
119864119899
119905119899
119899 (1)
Here is the list of the first Euler numbers
1198640= 1 119864
1= minus
1
2 119864
2= 0 119864
3=1
4
1198644= 0 119864
5= minus
1
2 119864
6= 0 119864
7=17
8
1198648= 0 119864
9= minus
31
2 119864
10= 0 119864
11=691
4
11986412= 0 119864
13= minus
5461
2 119864
14= 0
11986415=929569
16 119864
16= 0
11986417= minus
3202291
2 119864
18= 0
11986419=221930581
4 119864
20= 0
11986421= minus
4722116521
2 119864
22= 0
11986423=968383680827
8
11986424= 0
11986425= minus
14717667114151
2
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 568129 6 pageshttpdxdoiorg1011552014568129
2 Discrete Dynamics in Nature and Society
11986426= 0
11986427=2093660879252671
4
11986428= 0
(2)
The Euler polynomials 119864119899(119909) are defined by the generat-
ing function
119865 (119909 119905) =
infin
sum
119899=0
119864119899(119909)
119905119899
119899= (
2
119890119905 + 1) 119890119909119905
(3)
where we use the technique method notation by replacing119864(119909)119899 by 119864
Then it is easy to deduce that119864119896(119909) are polynomials of degree
119896 Here is the list of the first Euler polynomials
1198640(119909) = 1 119864
1(119909) = 119909 minus
1
2
1198642(119909) = 119909
2
minus 119909 1198643(119909) = 119909
3
minus31199092
2+1
4
1198644(119909) = 119909
4
minus 21199093
+ 119909 1198645(119909) = 119909
5
minus51199092
2minus1
2
1198646(119909) = 119909
6
minus 31199095
+ 51199093
minus 3119909
1198647(119909) = 119909
7
minus71199096
2+351199094
4minus211199092
2+17
8
1198648(119909) = 119909
8
minus 41199097
+ 141199095
minus 281199093
+ 17119909
1198649(119909) = 119909
9
minus91199098
2+ 21119909
6
minus 631199094
+1531199092
2minus31
2
11986410(119909) = 119909
10
minus 51199099
+ 301199097
minus 1261199095
+ 2551199093
minus 155119909
(6)
2 Generating Euler Polynomials and Numbers
Sinceinfin
sum
119899=0
119864119899(119909)
119905119899
119899=
2
119890119905 + 1119890119909119905
= 119890119864119905
119890119909119905
= 119890(119864+119909)119905
=
infin
sum
119899=0
(119864 + 119909)119899119905119899
119899
(7)
we have the following theorem
Theorem 1 For 119899 one has
119864119899(119909) =
119899
sum
119896=0
(119899
119896)119864119896119909119899minus119896
(8)
Definition 2 For 119904 isin C with Re(119904) gt 0 define the Euler zetafunction by
120577119864(119904) = 2
infin
sum
119899=1
(minus1)119899
119899119904 (9)
see [7ndash10]
Notice that the Euler zeta function can be analyticallycontinued to the whole complex plane and these zeta func-tions have the values of the Euler numbers at negative inte-gers That is Euler numbers are related to the Euler zetafunction as
120577119864(minus119899) = 119864
119899 (10)
Definition 3 We define the Hurwitz zeta function 120577119864(119904 119909) for
119904 isin C with Re(119904) gt 0 and 119909 isin R with 0 le 119909 lt 1 by
120577119864(119904 119909) = 2
infin
sum
119899=0
(minus1)119899
(119899 + 119909)119904 (11)
see [3 7ndash10 12]
Euler polynomials are related to the Hurwitz zeta func-tion as
120577119864(minus119899 119909) = 119864
119899(119909) (12)
We now consider the function119864(119904) as the analytic contin-uation of Euler numbers From the above analytic continua-tion of Euler numbers we consider
Γ (1 + 119896 + (119904 minus [119904])) Γ (1 + [119904] minus 119896)
=
[119904]+1
sum
119896=0
(minus1)119896minus1+[119904]
Γ (1 + 119904) 119864 ((119896 minus 1) + 119904 minus [119904]) 119908[119904]+1minus119896
Γ (119896 + (119904 minus [119904])) Γ (2 + [119904] minus 119896)
(22)
where [119904] gives the integer part of 119904 and so 119904minus[119904] gives the fra-ctional part
By (22) we obtain analytic continuation of Euler polyno-mials
1198640(119908) = minus1 119864
1(119908) = 119864 (1 119908) = minus05 + 119908
1198642(119908) = 119864 (2 119908) = 119908 minus 119908
2
4 Discrete Dynamics in Nature and Society
119864 (22 119908) asymp 00799175 + 0866594119908
minus 1197651541199082
+ 01161341199083
119864 (24 119908) asymp 0147412 + 0693349119908
minus 1361161199082
+ 02795031199083
119864 (26 119908) asymp 0199873 + 0485705119908
minus 1475471199082
+ 04869151199083
119864 (28 119908) asymp 0234806 + 0251271119908
minus 1526091199082
+ 07310151199083
1198643(119908) = 119864 (3 119908) = 025 minus 15119908
2
+ 1199083
(23)
By using (23) we plot the deformation of the curve119864(2 119908) into the curve of 119864(3 119908) via the real analytic contin-uation 119864(119904 119908) 2 le 119904 le 3 119908 isin R (see Figure 3)
Next we investigate the beautiful zeros of the 119864(119904 119908)by using a computer We plot the zeros of 119864(119904 119908) for 119904 =9 96 98 10 and 119908 isin C (Figure 4)
In Figure 4(a) we choose 119904 = 9 In Figure 4(b) we choose119904 = 96 In Figure 4(c) we choose 119904 = 98 In Figure 4(d) wechoose 119904 = 10
Theorem 4 If 119899 equiv 1 (mod 2) then 119864119899(12) = 0 for 119899 isin N
The question is what happens with the reflexive symme-try (25) when one considers Euler polynomials Prove that119864119899(119909) 119909 isin C has Re(119909) = 12 reflection symmetry in addi-
tion to the usual Im(119909) = 0 reflection symmetry analyticcomplex functions However we observe that 119864(119904 119908)119908 isin Cdoes not have Re(119908) = 12 reflection symmetry analyticcomplex functions (Figure 4)
Our numerical results for approximate solutions of realzeros of 119864(119904 119908) are displayed We observe a remarkably reg-ular structure of the complex roots of Euler polynomials Wehope to verify a remarkably regular structure of the complexroots of Euler polynomials (Table 1) Next we calculated anapproximate solution satisfying 119864(119904 119908) 119908 isin R The resultsare given in Table 2
0
minus05 minus04 minus03 minus02 minus01 0 01
02
minus08
minus06
minus04
minus02
w
E(sw
)
E(3 w)
E(2 w)
Figure 3 The curve of 119864(119904 119908) 2 le 119904 le 3 minus05 le 119908 le 02
Table 1 Numbers of real and complex zeros of 119864(119904 119908)
Figure 4 Zeros of 119864(119904 119908) for 119904 = 9 96 98 10
Euler polynomials 119864119899(119908) are polynomials of degree 119899
Thus 119864119899(119908) has 119899 zeros and 119864
119899+1(119908) has 119899 + 1 zeros When
discrete 119899 is analytically continued to continuous parameter 119904it naturally leads to the following question how does 119864(119904 119908)the analytic continuation of119864
119899(119908) pick up an additional zero
as 119904 increases continuously by oneThis introduces the exciting concept of the dynamics
of the zeros of analytic continued polynomials the idea oflooking at how the zeros move about in the 119908 complex plane
as we vary the parameter 119904 For more studies and results inthis subject you may see [11 14ndash16]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
6 Discrete Dynamics in Nature and Society
References
[1] S Araci and M Acikgoz ldquoA note on the Frobenius-Eulernumbers and polynomials associated with Bernstein polynomi-alsrdquoAdvanced Studies inContemporaryMathematics vol 22 pp399ndash406 2012
[2] R Ayoub ldquoEuler zeta functionrdquo The American MathematicalMonthly vol 81 pp 1067ndash1086 1974
[3] A Bayad ldquoModular properties of elliptic Bernoulli and EulerfunctionsrdquoAdvanced Studies inContemporaryMathematics vol20 no 3 pp 389ndash401 2010
[4] I N Cangul H Ozden and Y Simsek ldquoA new approach to q-Genocchi numbers and their interpolation functionsrdquo Nonlin-ear Analysis Theory Methods and Applications vol 71 no 12pp e793ndashe799 2009
[5] L-C Jang ldquoOn multiple generalized 120596-Genocchi polynomialsand their applicationsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 316870 8 pages 2010
[6] J Y Kang H Y Lee and N S Jung ldquoSome relations of thetwisted q-Genocchi numbers and polynomials with weight 120572and weak Weight 120573rdquo Abstract and Applied Analysis vol 2012Article ID 860921 9 pages 2012
[7] M S Kim and S Hu ldquoOn p-adic Hurwitz-type Euler Zeta func-tionsrdquo Journal of Number Theory no 132 pp 2977ndash3015 2012
[8] T Kim ldquoBarnes-type multiple q-zeta functions and q-Eulerpolynomialsrdquo Journal of Physics A Mathematical and Theoret-ical vol 43 Article ID 255201 11 pages 2010
[9] T Kim ldquoNote on the Euler q-zeta functionsrdquo Journal of NumberTheory vol 129 no 7 pp 1798ndash1804 2009
[10] T Kim ldquoOn p-adic interpolating function for q-Euler numbersand its derivativesrdquo Journal of Mathematical Analysis and Appli-cations vol 339 no 1 pp 598ndash608 2008
[11] T Kim C S Ryoo L C Jang and S H Rim ldquoExploring the q-riemann zeta function and q-bernoulli polynomialsrdquo DiscreteDynamics in Nature and Society vol 2005 no 2 pp 171ndash1812005
[12] H Ozden and Y Simsek ldquoA new extension of q-Euler num-bers and polynomials related to their interpolation functionsrdquoApplied Mathematics Letters vol 21 no 9 pp 934ndash939 2008
[13] K H Park S-H Rim and E J Moon ldquoOn Genocchi numbersandpolynomialsrdquoAbstract andAppliedAnalysis vol 2008Arti-cle ID 898471 7 pages 2008
[14] C S Ryoo ldquoA numerical computation on the structure of theroots of q-extension of Genocchi polynomialsrdquo Applied Mathe-matics Letters vol 21 no 4 pp 348ndash354 2008
[15] C S Ryoo ldquoCalculating zeros of the second kind Euler polyno-mialsrdquo Journal of Computational Analysis and Applications vol12 pp 828ndash833 2010
[16] C S Ryoo T Kim and R P Agarwal ldquoA numerical investi-gation of the roots of q-polynomialsrdquo International Journal ofComputer Mathematics vol 83 no 2 pp 223ndash234 2006
[17] Y Simsek ldquoGenerating functions of the twisted Bernoulli Num-bers and polynomials associated with their interpolation func-tionsrdquo Advanced Studies in Contemporary Mathematics vol 16no 2 pp 251ndash257 2008
[18] Y Simsek ldquoTwisted (h q)-Bernoulli numbers and polynomialsrelated to twisted (h q)-zeta function andL-functionrdquo Journal ofMathematical Analysis and Applications vol 324 no 2 pp 790ndash804 2006
[19] Y Simsek V Kurt and D Kim ldquoNew approach to the completesum of products of the twisted (h q)-Bernoulli numbers andpolynomialsrdquo Journal ofNonlinearMathematical Physics vol 14no 1 pp 44ndash56 2007
Then it is easy to deduce that119864119896(119909) are polynomials of degree
119896 Here is the list of the first Euler polynomials
1198640(119909) = 1 119864
1(119909) = 119909 minus
1
2
1198642(119909) = 119909
2
minus 119909 1198643(119909) = 119909
3
minus31199092
2+1
4
1198644(119909) = 119909
4
minus 21199093
+ 119909 1198645(119909) = 119909
5
minus51199092
2minus1
2
1198646(119909) = 119909
6
minus 31199095
+ 51199093
minus 3119909
1198647(119909) = 119909
7
minus71199096
2+351199094
4minus211199092
2+17
8
1198648(119909) = 119909
8
minus 41199097
+ 141199095
minus 281199093
+ 17119909
1198649(119909) = 119909
9
minus91199098
2+ 21119909
6
minus 631199094
+1531199092
2minus31
2
11986410(119909) = 119909
10
minus 51199099
+ 301199097
minus 1261199095
+ 2551199093
minus 155119909
(6)
2 Generating Euler Polynomials and Numbers
Sinceinfin
sum
119899=0
119864119899(119909)
119905119899
119899=
2
119890119905 + 1119890119909119905
= 119890119864119905
119890119909119905
= 119890(119864+119909)119905
=
infin
sum
119899=0
(119864 + 119909)119899119905119899
119899
(7)
we have the following theorem
Theorem 1 For 119899 one has
119864119899(119909) =
119899
sum
119896=0
(119899
119896)119864119896119909119899minus119896
(8)
Definition 2 For 119904 isin C with Re(119904) gt 0 define the Euler zetafunction by
120577119864(119904) = 2
infin
sum
119899=1
(minus1)119899
119899119904 (9)
see [7ndash10]
Notice that the Euler zeta function can be analyticallycontinued to the whole complex plane and these zeta func-tions have the values of the Euler numbers at negative inte-gers That is Euler numbers are related to the Euler zetafunction as
120577119864(minus119899) = 119864
119899 (10)
Definition 3 We define the Hurwitz zeta function 120577119864(119904 119909) for
119904 isin C with Re(119904) gt 0 and 119909 isin R with 0 le 119909 lt 1 by
120577119864(119904 119909) = 2
infin
sum
119899=0
(minus1)119899
(119899 + 119909)119904 (11)
see [3 7ndash10 12]
Euler polynomials are related to the Hurwitz zeta func-tion as
120577119864(minus119899 119909) = 119864
119899(119909) (12)
We now consider the function119864(119904) as the analytic contin-uation of Euler numbers From the above analytic continua-tion of Euler numbers we consider
Γ (1 + 119896 + (119904 minus [119904])) Γ (1 + [119904] minus 119896)
=
[119904]+1
sum
119896=0
(minus1)119896minus1+[119904]
Γ (1 + 119904) 119864 ((119896 minus 1) + 119904 minus [119904]) 119908[119904]+1minus119896
Γ (119896 + (119904 minus [119904])) Γ (2 + [119904] minus 119896)
(22)
where [119904] gives the integer part of 119904 and so 119904minus[119904] gives the fra-ctional part
By (22) we obtain analytic continuation of Euler polyno-mials
1198640(119908) = minus1 119864
1(119908) = 119864 (1 119908) = minus05 + 119908
1198642(119908) = 119864 (2 119908) = 119908 minus 119908
2
4 Discrete Dynamics in Nature and Society
119864 (22 119908) asymp 00799175 + 0866594119908
minus 1197651541199082
+ 01161341199083
119864 (24 119908) asymp 0147412 + 0693349119908
minus 1361161199082
+ 02795031199083
119864 (26 119908) asymp 0199873 + 0485705119908
minus 1475471199082
+ 04869151199083
119864 (28 119908) asymp 0234806 + 0251271119908
minus 1526091199082
+ 07310151199083
1198643(119908) = 119864 (3 119908) = 025 minus 15119908
2
+ 1199083
(23)
By using (23) we plot the deformation of the curve119864(2 119908) into the curve of 119864(3 119908) via the real analytic contin-uation 119864(119904 119908) 2 le 119904 le 3 119908 isin R (see Figure 3)
Next we investigate the beautiful zeros of the 119864(119904 119908)by using a computer We plot the zeros of 119864(119904 119908) for 119904 =9 96 98 10 and 119908 isin C (Figure 4)
In Figure 4(a) we choose 119904 = 9 In Figure 4(b) we choose119904 = 96 In Figure 4(c) we choose 119904 = 98 In Figure 4(d) wechoose 119904 = 10
Theorem 4 If 119899 equiv 1 (mod 2) then 119864119899(12) = 0 for 119899 isin N
The question is what happens with the reflexive symme-try (25) when one considers Euler polynomials Prove that119864119899(119909) 119909 isin C has Re(119909) = 12 reflection symmetry in addi-
tion to the usual Im(119909) = 0 reflection symmetry analyticcomplex functions However we observe that 119864(119904 119908)119908 isin Cdoes not have Re(119908) = 12 reflection symmetry analyticcomplex functions (Figure 4)
Our numerical results for approximate solutions of realzeros of 119864(119904 119908) are displayed We observe a remarkably reg-ular structure of the complex roots of Euler polynomials Wehope to verify a remarkably regular structure of the complexroots of Euler polynomials (Table 1) Next we calculated anapproximate solution satisfying 119864(119904 119908) 119908 isin R The resultsare given in Table 2
0
minus05 minus04 minus03 minus02 minus01 0 01
02
minus08
minus06
minus04
minus02
w
E(sw
)
E(3 w)
E(2 w)
Figure 3 The curve of 119864(119904 119908) 2 le 119904 le 3 minus05 le 119908 le 02
Table 1 Numbers of real and complex zeros of 119864(119904 119908)
Figure 4 Zeros of 119864(119904 119908) for 119904 = 9 96 98 10
Euler polynomials 119864119899(119908) are polynomials of degree 119899
Thus 119864119899(119908) has 119899 zeros and 119864
119899+1(119908) has 119899 + 1 zeros When
discrete 119899 is analytically continued to continuous parameter 119904it naturally leads to the following question how does 119864(119904 119908)the analytic continuation of119864
119899(119908) pick up an additional zero
as 119904 increases continuously by oneThis introduces the exciting concept of the dynamics
of the zeros of analytic continued polynomials the idea oflooking at how the zeros move about in the 119908 complex plane
as we vary the parameter 119904 For more studies and results inthis subject you may see [11 14ndash16]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
6 Discrete Dynamics in Nature and Society
References
[1] S Araci and M Acikgoz ldquoA note on the Frobenius-Eulernumbers and polynomials associated with Bernstein polynomi-alsrdquoAdvanced Studies inContemporaryMathematics vol 22 pp399ndash406 2012
[2] R Ayoub ldquoEuler zeta functionrdquo The American MathematicalMonthly vol 81 pp 1067ndash1086 1974
[3] A Bayad ldquoModular properties of elliptic Bernoulli and EulerfunctionsrdquoAdvanced Studies inContemporaryMathematics vol20 no 3 pp 389ndash401 2010
[4] I N Cangul H Ozden and Y Simsek ldquoA new approach to q-Genocchi numbers and their interpolation functionsrdquo Nonlin-ear Analysis Theory Methods and Applications vol 71 no 12pp e793ndashe799 2009
[5] L-C Jang ldquoOn multiple generalized 120596-Genocchi polynomialsand their applicationsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 316870 8 pages 2010
[6] J Y Kang H Y Lee and N S Jung ldquoSome relations of thetwisted q-Genocchi numbers and polynomials with weight 120572and weak Weight 120573rdquo Abstract and Applied Analysis vol 2012Article ID 860921 9 pages 2012
[7] M S Kim and S Hu ldquoOn p-adic Hurwitz-type Euler Zeta func-tionsrdquo Journal of Number Theory no 132 pp 2977ndash3015 2012
[8] T Kim ldquoBarnes-type multiple q-zeta functions and q-Eulerpolynomialsrdquo Journal of Physics A Mathematical and Theoret-ical vol 43 Article ID 255201 11 pages 2010
[9] T Kim ldquoNote on the Euler q-zeta functionsrdquo Journal of NumberTheory vol 129 no 7 pp 1798ndash1804 2009
[10] T Kim ldquoOn p-adic interpolating function for q-Euler numbersand its derivativesrdquo Journal of Mathematical Analysis and Appli-cations vol 339 no 1 pp 598ndash608 2008
[11] T Kim C S Ryoo L C Jang and S H Rim ldquoExploring the q-riemann zeta function and q-bernoulli polynomialsrdquo DiscreteDynamics in Nature and Society vol 2005 no 2 pp 171ndash1812005
[12] H Ozden and Y Simsek ldquoA new extension of q-Euler num-bers and polynomials related to their interpolation functionsrdquoApplied Mathematics Letters vol 21 no 9 pp 934ndash939 2008
[13] K H Park S-H Rim and E J Moon ldquoOn Genocchi numbersandpolynomialsrdquoAbstract andAppliedAnalysis vol 2008Arti-cle ID 898471 7 pages 2008
[14] C S Ryoo ldquoA numerical computation on the structure of theroots of q-extension of Genocchi polynomialsrdquo Applied Mathe-matics Letters vol 21 no 4 pp 348ndash354 2008
[15] C S Ryoo ldquoCalculating zeros of the second kind Euler polyno-mialsrdquo Journal of Computational Analysis and Applications vol12 pp 828ndash833 2010
[16] C S Ryoo T Kim and R P Agarwal ldquoA numerical investi-gation of the roots of q-polynomialsrdquo International Journal ofComputer Mathematics vol 83 no 2 pp 223ndash234 2006
[17] Y Simsek ldquoGenerating functions of the twisted Bernoulli Num-bers and polynomials associated with their interpolation func-tionsrdquo Advanced Studies in Contemporary Mathematics vol 16no 2 pp 251ndash257 2008
[18] Y Simsek ldquoTwisted (h q)-Bernoulli numbers and polynomialsrelated to twisted (h q)-zeta function andL-functionrdquo Journal ofMathematical Analysis and Applications vol 324 no 2 pp 790ndash804 2006
[19] Y Simsek V Kurt and D Kim ldquoNew approach to the completesum of products of the twisted (h q)-Bernoulli numbers andpolynomialsrdquo Journal ofNonlinearMathematical Physics vol 14no 1 pp 44ndash56 2007
Γ (1 + 119896 + (119904 minus [119904])) Γ (1 + [119904] minus 119896)
=
[119904]+1
sum
119896=0
(minus1)119896minus1+[119904]
Γ (1 + 119904) 119864 ((119896 minus 1) + 119904 minus [119904]) 119908[119904]+1minus119896
Γ (119896 + (119904 minus [119904])) Γ (2 + [119904] minus 119896)
(22)
where [119904] gives the integer part of 119904 and so 119904minus[119904] gives the fra-ctional part
By (22) we obtain analytic continuation of Euler polyno-mials
1198640(119908) = minus1 119864
1(119908) = 119864 (1 119908) = minus05 + 119908
1198642(119908) = 119864 (2 119908) = 119908 minus 119908
2
4 Discrete Dynamics in Nature and Society
119864 (22 119908) asymp 00799175 + 0866594119908
minus 1197651541199082
+ 01161341199083
119864 (24 119908) asymp 0147412 + 0693349119908
minus 1361161199082
+ 02795031199083
119864 (26 119908) asymp 0199873 + 0485705119908
minus 1475471199082
+ 04869151199083
119864 (28 119908) asymp 0234806 + 0251271119908
minus 1526091199082
+ 07310151199083
1198643(119908) = 119864 (3 119908) = 025 minus 15119908
2
+ 1199083
(23)
By using (23) we plot the deformation of the curve119864(2 119908) into the curve of 119864(3 119908) via the real analytic contin-uation 119864(119904 119908) 2 le 119904 le 3 119908 isin R (see Figure 3)
Next we investigate the beautiful zeros of the 119864(119904 119908)by using a computer We plot the zeros of 119864(119904 119908) for 119904 =9 96 98 10 and 119908 isin C (Figure 4)
In Figure 4(a) we choose 119904 = 9 In Figure 4(b) we choose119904 = 96 In Figure 4(c) we choose 119904 = 98 In Figure 4(d) wechoose 119904 = 10
Theorem 4 If 119899 equiv 1 (mod 2) then 119864119899(12) = 0 for 119899 isin N
The question is what happens with the reflexive symme-try (25) when one considers Euler polynomials Prove that119864119899(119909) 119909 isin C has Re(119909) = 12 reflection symmetry in addi-
tion to the usual Im(119909) = 0 reflection symmetry analyticcomplex functions However we observe that 119864(119904 119908)119908 isin Cdoes not have Re(119908) = 12 reflection symmetry analyticcomplex functions (Figure 4)
Our numerical results for approximate solutions of realzeros of 119864(119904 119908) are displayed We observe a remarkably reg-ular structure of the complex roots of Euler polynomials Wehope to verify a remarkably regular structure of the complexroots of Euler polynomials (Table 1) Next we calculated anapproximate solution satisfying 119864(119904 119908) 119908 isin R The resultsare given in Table 2
0
minus05 minus04 minus03 minus02 minus01 0 01
02
minus08
minus06
minus04
minus02
w
E(sw
)
E(3 w)
E(2 w)
Figure 3 The curve of 119864(119904 119908) 2 le 119904 le 3 minus05 le 119908 le 02
Table 1 Numbers of real and complex zeros of 119864(119904 119908)
Figure 4 Zeros of 119864(119904 119908) for 119904 = 9 96 98 10
Euler polynomials 119864119899(119908) are polynomials of degree 119899
Thus 119864119899(119908) has 119899 zeros and 119864
119899+1(119908) has 119899 + 1 zeros When
discrete 119899 is analytically continued to continuous parameter 119904it naturally leads to the following question how does 119864(119904 119908)the analytic continuation of119864
119899(119908) pick up an additional zero
as 119904 increases continuously by oneThis introduces the exciting concept of the dynamics
of the zeros of analytic continued polynomials the idea oflooking at how the zeros move about in the 119908 complex plane
as we vary the parameter 119904 For more studies and results inthis subject you may see [11 14ndash16]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
6 Discrete Dynamics in Nature and Society
References
[1] S Araci and M Acikgoz ldquoA note on the Frobenius-Eulernumbers and polynomials associated with Bernstein polynomi-alsrdquoAdvanced Studies inContemporaryMathematics vol 22 pp399ndash406 2012
[2] R Ayoub ldquoEuler zeta functionrdquo The American MathematicalMonthly vol 81 pp 1067ndash1086 1974
[3] A Bayad ldquoModular properties of elliptic Bernoulli and EulerfunctionsrdquoAdvanced Studies inContemporaryMathematics vol20 no 3 pp 389ndash401 2010
[4] I N Cangul H Ozden and Y Simsek ldquoA new approach to q-Genocchi numbers and their interpolation functionsrdquo Nonlin-ear Analysis Theory Methods and Applications vol 71 no 12pp e793ndashe799 2009
[5] L-C Jang ldquoOn multiple generalized 120596-Genocchi polynomialsand their applicationsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 316870 8 pages 2010
[6] J Y Kang H Y Lee and N S Jung ldquoSome relations of thetwisted q-Genocchi numbers and polynomials with weight 120572and weak Weight 120573rdquo Abstract and Applied Analysis vol 2012Article ID 860921 9 pages 2012
[7] M S Kim and S Hu ldquoOn p-adic Hurwitz-type Euler Zeta func-tionsrdquo Journal of Number Theory no 132 pp 2977ndash3015 2012
[8] T Kim ldquoBarnes-type multiple q-zeta functions and q-Eulerpolynomialsrdquo Journal of Physics A Mathematical and Theoret-ical vol 43 Article ID 255201 11 pages 2010
[9] T Kim ldquoNote on the Euler q-zeta functionsrdquo Journal of NumberTheory vol 129 no 7 pp 1798ndash1804 2009
[10] T Kim ldquoOn p-adic interpolating function for q-Euler numbersand its derivativesrdquo Journal of Mathematical Analysis and Appli-cations vol 339 no 1 pp 598ndash608 2008
[11] T Kim C S Ryoo L C Jang and S H Rim ldquoExploring the q-riemann zeta function and q-bernoulli polynomialsrdquo DiscreteDynamics in Nature and Society vol 2005 no 2 pp 171ndash1812005
[12] H Ozden and Y Simsek ldquoA new extension of q-Euler num-bers and polynomials related to their interpolation functionsrdquoApplied Mathematics Letters vol 21 no 9 pp 934ndash939 2008
[13] K H Park S-H Rim and E J Moon ldquoOn Genocchi numbersandpolynomialsrdquoAbstract andAppliedAnalysis vol 2008Arti-cle ID 898471 7 pages 2008
[14] C S Ryoo ldquoA numerical computation on the structure of theroots of q-extension of Genocchi polynomialsrdquo Applied Mathe-matics Letters vol 21 no 4 pp 348ndash354 2008
[15] C S Ryoo ldquoCalculating zeros of the second kind Euler polyno-mialsrdquo Journal of Computational Analysis and Applications vol12 pp 828ndash833 2010
[16] C S Ryoo T Kim and R P Agarwal ldquoA numerical investi-gation of the roots of q-polynomialsrdquo International Journal ofComputer Mathematics vol 83 no 2 pp 223ndash234 2006
[17] Y Simsek ldquoGenerating functions of the twisted Bernoulli Num-bers and polynomials associated with their interpolation func-tionsrdquo Advanced Studies in Contemporary Mathematics vol 16no 2 pp 251ndash257 2008
[18] Y Simsek ldquoTwisted (h q)-Bernoulli numbers and polynomialsrelated to twisted (h q)-zeta function andL-functionrdquo Journal ofMathematical Analysis and Applications vol 324 no 2 pp 790ndash804 2006
[19] Y Simsek V Kurt and D Kim ldquoNew approach to the completesum of products of the twisted (h q)-Bernoulli numbers andpolynomialsrdquo Journal ofNonlinearMathematical Physics vol 14no 1 pp 44ndash56 2007
1198643(119908) = 119864 (3 119908) = 025 minus 15119908
2
+ 1199083
(23)
By using (23) we plot the deformation of the curve119864(2 119908) into the curve of 119864(3 119908) via the real analytic contin-uation 119864(119904 119908) 2 le 119904 le 3 119908 isin R (see Figure 3)
Next we investigate the beautiful zeros of the 119864(119904 119908)by using a computer We plot the zeros of 119864(119904 119908) for 119904 =9 96 98 10 and 119908 isin C (Figure 4)
In Figure 4(a) we choose 119904 = 9 In Figure 4(b) we choose119904 = 96 In Figure 4(c) we choose 119904 = 98 In Figure 4(d) wechoose 119904 = 10
Theorem 4 If 119899 equiv 1 (mod 2) then 119864119899(12) = 0 for 119899 isin N
The question is what happens with the reflexive symme-try (25) when one considers Euler polynomials Prove that119864119899(119909) 119909 isin C has Re(119909) = 12 reflection symmetry in addi-
tion to the usual Im(119909) = 0 reflection symmetry analyticcomplex functions However we observe that 119864(119904 119908)119908 isin Cdoes not have Re(119908) = 12 reflection symmetry analyticcomplex functions (Figure 4)
Our numerical results for approximate solutions of realzeros of 119864(119904 119908) are displayed We observe a remarkably reg-ular structure of the complex roots of Euler polynomials Wehope to verify a remarkably regular structure of the complexroots of Euler polynomials (Table 1) Next we calculated anapproximate solution satisfying 119864(119904 119908) 119908 isin R The resultsare given in Table 2
0
minus05 minus04 minus03 minus02 minus01 0 01
02
minus08
minus06
minus04
minus02
w
E(sw
)
E(3 w)
E(2 w)
Figure 3 The curve of 119864(119904 119908) 2 le 119904 le 3 minus05 le 119908 le 02
Table 1 Numbers of real and complex zeros of 119864(119904 119908)
Figure 4 Zeros of 119864(119904 119908) for 119904 = 9 96 98 10
Euler polynomials 119864119899(119908) are polynomials of degree 119899
Thus 119864119899(119908) has 119899 zeros and 119864
119899+1(119908) has 119899 + 1 zeros When
discrete 119899 is analytically continued to continuous parameter 119904it naturally leads to the following question how does 119864(119904 119908)the analytic continuation of119864
119899(119908) pick up an additional zero
as 119904 increases continuously by oneThis introduces the exciting concept of the dynamics
of the zeros of analytic continued polynomials the idea oflooking at how the zeros move about in the 119908 complex plane
as we vary the parameter 119904 For more studies and results inthis subject you may see [11 14ndash16]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
6 Discrete Dynamics in Nature and Society
References
[1] S Araci and M Acikgoz ldquoA note on the Frobenius-Eulernumbers and polynomials associated with Bernstein polynomi-alsrdquoAdvanced Studies inContemporaryMathematics vol 22 pp399ndash406 2012
[2] R Ayoub ldquoEuler zeta functionrdquo The American MathematicalMonthly vol 81 pp 1067ndash1086 1974
[3] A Bayad ldquoModular properties of elliptic Bernoulli and EulerfunctionsrdquoAdvanced Studies inContemporaryMathematics vol20 no 3 pp 389ndash401 2010
[4] I N Cangul H Ozden and Y Simsek ldquoA new approach to q-Genocchi numbers and their interpolation functionsrdquo Nonlin-ear Analysis Theory Methods and Applications vol 71 no 12pp e793ndashe799 2009
[5] L-C Jang ldquoOn multiple generalized 120596-Genocchi polynomialsand their applicationsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 316870 8 pages 2010
[6] J Y Kang H Y Lee and N S Jung ldquoSome relations of thetwisted q-Genocchi numbers and polynomials with weight 120572and weak Weight 120573rdquo Abstract and Applied Analysis vol 2012Article ID 860921 9 pages 2012
[7] M S Kim and S Hu ldquoOn p-adic Hurwitz-type Euler Zeta func-tionsrdquo Journal of Number Theory no 132 pp 2977ndash3015 2012
[8] T Kim ldquoBarnes-type multiple q-zeta functions and q-Eulerpolynomialsrdquo Journal of Physics A Mathematical and Theoret-ical vol 43 Article ID 255201 11 pages 2010
[9] T Kim ldquoNote on the Euler q-zeta functionsrdquo Journal of NumberTheory vol 129 no 7 pp 1798ndash1804 2009
[10] T Kim ldquoOn p-adic interpolating function for q-Euler numbersand its derivativesrdquo Journal of Mathematical Analysis and Appli-cations vol 339 no 1 pp 598ndash608 2008
[11] T Kim C S Ryoo L C Jang and S H Rim ldquoExploring the q-riemann zeta function and q-bernoulli polynomialsrdquo DiscreteDynamics in Nature and Society vol 2005 no 2 pp 171ndash1812005
[12] H Ozden and Y Simsek ldquoA new extension of q-Euler num-bers and polynomials related to their interpolation functionsrdquoApplied Mathematics Letters vol 21 no 9 pp 934ndash939 2008
[13] K H Park S-H Rim and E J Moon ldquoOn Genocchi numbersandpolynomialsrdquoAbstract andAppliedAnalysis vol 2008Arti-cle ID 898471 7 pages 2008
[14] C S Ryoo ldquoA numerical computation on the structure of theroots of q-extension of Genocchi polynomialsrdquo Applied Mathe-matics Letters vol 21 no 4 pp 348ndash354 2008
[15] C S Ryoo ldquoCalculating zeros of the second kind Euler polyno-mialsrdquo Journal of Computational Analysis and Applications vol12 pp 828ndash833 2010
[16] C S Ryoo T Kim and R P Agarwal ldquoA numerical investi-gation of the roots of q-polynomialsrdquo International Journal ofComputer Mathematics vol 83 no 2 pp 223ndash234 2006
[17] Y Simsek ldquoGenerating functions of the twisted Bernoulli Num-bers and polynomials associated with their interpolation func-tionsrdquo Advanced Studies in Contemporary Mathematics vol 16no 2 pp 251ndash257 2008
[18] Y Simsek ldquoTwisted (h q)-Bernoulli numbers and polynomialsrelated to twisted (h q)-zeta function andL-functionrdquo Journal ofMathematical Analysis and Applications vol 324 no 2 pp 790ndash804 2006
[19] Y Simsek V Kurt and D Kim ldquoNew approach to the completesum of products of the twisted (h q)-Bernoulli numbers andpolynomialsrdquo Journal ofNonlinearMathematical Physics vol 14no 1 pp 44ndash56 2007
Figure 4 Zeros of 119864(119904 119908) for 119904 = 9 96 98 10
Euler polynomials 119864119899(119908) are polynomials of degree 119899
Thus 119864119899(119908) has 119899 zeros and 119864
119899+1(119908) has 119899 + 1 zeros When
discrete 119899 is analytically continued to continuous parameter 119904it naturally leads to the following question how does 119864(119904 119908)the analytic continuation of119864
119899(119908) pick up an additional zero
as 119904 increases continuously by oneThis introduces the exciting concept of the dynamics
of the zeros of analytic continued polynomials the idea oflooking at how the zeros move about in the 119908 complex plane
as we vary the parameter 119904 For more studies and results inthis subject you may see [11 14ndash16]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
6 Discrete Dynamics in Nature and Society
References
[1] S Araci and M Acikgoz ldquoA note on the Frobenius-Eulernumbers and polynomials associated with Bernstein polynomi-alsrdquoAdvanced Studies inContemporaryMathematics vol 22 pp399ndash406 2012
[2] R Ayoub ldquoEuler zeta functionrdquo The American MathematicalMonthly vol 81 pp 1067ndash1086 1974
[3] A Bayad ldquoModular properties of elliptic Bernoulli and EulerfunctionsrdquoAdvanced Studies inContemporaryMathematics vol20 no 3 pp 389ndash401 2010
[4] I N Cangul H Ozden and Y Simsek ldquoA new approach to q-Genocchi numbers and their interpolation functionsrdquo Nonlin-ear Analysis Theory Methods and Applications vol 71 no 12pp e793ndashe799 2009
[5] L-C Jang ldquoOn multiple generalized 120596-Genocchi polynomialsand their applicationsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 316870 8 pages 2010
[6] J Y Kang H Y Lee and N S Jung ldquoSome relations of thetwisted q-Genocchi numbers and polynomials with weight 120572and weak Weight 120573rdquo Abstract and Applied Analysis vol 2012Article ID 860921 9 pages 2012
[7] M S Kim and S Hu ldquoOn p-adic Hurwitz-type Euler Zeta func-tionsrdquo Journal of Number Theory no 132 pp 2977ndash3015 2012
[8] T Kim ldquoBarnes-type multiple q-zeta functions and q-Eulerpolynomialsrdquo Journal of Physics A Mathematical and Theoret-ical vol 43 Article ID 255201 11 pages 2010
[9] T Kim ldquoNote on the Euler q-zeta functionsrdquo Journal of NumberTheory vol 129 no 7 pp 1798ndash1804 2009
[10] T Kim ldquoOn p-adic interpolating function for q-Euler numbersand its derivativesrdquo Journal of Mathematical Analysis and Appli-cations vol 339 no 1 pp 598ndash608 2008
[11] T Kim C S Ryoo L C Jang and S H Rim ldquoExploring the q-riemann zeta function and q-bernoulli polynomialsrdquo DiscreteDynamics in Nature and Society vol 2005 no 2 pp 171ndash1812005
[12] H Ozden and Y Simsek ldquoA new extension of q-Euler num-bers and polynomials related to their interpolation functionsrdquoApplied Mathematics Letters vol 21 no 9 pp 934ndash939 2008
[13] K H Park S-H Rim and E J Moon ldquoOn Genocchi numbersandpolynomialsrdquoAbstract andAppliedAnalysis vol 2008Arti-cle ID 898471 7 pages 2008
[14] C S Ryoo ldquoA numerical computation on the structure of theroots of q-extension of Genocchi polynomialsrdquo Applied Mathe-matics Letters vol 21 no 4 pp 348ndash354 2008
[15] C S Ryoo ldquoCalculating zeros of the second kind Euler polyno-mialsrdquo Journal of Computational Analysis and Applications vol12 pp 828ndash833 2010
[16] C S Ryoo T Kim and R P Agarwal ldquoA numerical investi-gation of the roots of q-polynomialsrdquo International Journal ofComputer Mathematics vol 83 no 2 pp 223ndash234 2006
[17] Y Simsek ldquoGenerating functions of the twisted Bernoulli Num-bers and polynomials associated with their interpolation func-tionsrdquo Advanced Studies in Contemporary Mathematics vol 16no 2 pp 251ndash257 2008
[18] Y Simsek ldquoTwisted (h q)-Bernoulli numbers and polynomialsrelated to twisted (h q)-zeta function andL-functionrdquo Journal ofMathematical Analysis and Applications vol 324 no 2 pp 790ndash804 2006
[19] Y Simsek V Kurt and D Kim ldquoNew approach to the completesum of products of the twisted (h q)-Bernoulli numbers andpolynomialsrdquo Journal ofNonlinearMathematical Physics vol 14no 1 pp 44ndash56 2007
[1] S Araci and M Acikgoz ldquoA note on the Frobenius-Eulernumbers and polynomials associated with Bernstein polynomi-alsrdquoAdvanced Studies inContemporaryMathematics vol 22 pp399ndash406 2012
[2] R Ayoub ldquoEuler zeta functionrdquo The American MathematicalMonthly vol 81 pp 1067ndash1086 1974
[3] A Bayad ldquoModular properties of elliptic Bernoulli and EulerfunctionsrdquoAdvanced Studies inContemporaryMathematics vol20 no 3 pp 389ndash401 2010
[4] I N Cangul H Ozden and Y Simsek ldquoA new approach to q-Genocchi numbers and their interpolation functionsrdquo Nonlin-ear Analysis Theory Methods and Applications vol 71 no 12pp e793ndashe799 2009
[5] L-C Jang ldquoOn multiple generalized 120596-Genocchi polynomialsand their applicationsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 316870 8 pages 2010
[6] J Y Kang H Y Lee and N S Jung ldquoSome relations of thetwisted q-Genocchi numbers and polynomials with weight 120572and weak Weight 120573rdquo Abstract and Applied Analysis vol 2012Article ID 860921 9 pages 2012
[7] M S Kim and S Hu ldquoOn p-adic Hurwitz-type Euler Zeta func-tionsrdquo Journal of Number Theory no 132 pp 2977ndash3015 2012
[8] T Kim ldquoBarnes-type multiple q-zeta functions and q-Eulerpolynomialsrdquo Journal of Physics A Mathematical and Theoret-ical vol 43 Article ID 255201 11 pages 2010
[9] T Kim ldquoNote on the Euler q-zeta functionsrdquo Journal of NumberTheory vol 129 no 7 pp 1798ndash1804 2009
[10] T Kim ldquoOn p-adic interpolating function for q-Euler numbersand its derivativesrdquo Journal of Mathematical Analysis and Appli-cations vol 339 no 1 pp 598ndash608 2008
[11] T Kim C S Ryoo L C Jang and S H Rim ldquoExploring the q-riemann zeta function and q-bernoulli polynomialsrdquo DiscreteDynamics in Nature and Society vol 2005 no 2 pp 171ndash1812005
[12] H Ozden and Y Simsek ldquoA new extension of q-Euler num-bers and polynomials related to their interpolation functionsrdquoApplied Mathematics Letters vol 21 no 9 pp 934ndash939 2008
[13] K H Park S-H Rim and E J Moon ldquoOn Genocchi numbersandpolynomialsrdquoAbstract andAppliedAnalysis vol 2008Arti-cle ID 898471 7 pages 2008
[14] C S Ryoo ldquoA numerical computation on the structure of theroots of q-extension of Genocchi polynomialsrdquo Applied Mathe-matics Letters vol 21 no 4 pp 348ndash354 2008
[15] C S Ryoo ldquoCalculating zeros of the second kind Euler polyno-mialsrdquo Journal of Computational Analysis and Applications vol12 pp 828ndash833 2010
[16] C S Ryoo T Kim and R P Agarwal ldquoA numerical investi-gation of the roots of q-polynomialsrdquo International Journal ofComputer Mathematics vol 83 no 2 pp 223ndash234 2006
[17] Y Simsek ldquoGenerating functions of the twisted Bernoulli Num-bers and polynomials associated with their interpolation func-tionsrdquo Advanced Studies in Contemporary Mathematics vol 16no 2 pp 251ndash257 2008
[18] Y Simsek ldquoTwisted (h q)-Bernoulli numbers and polynomialsrelated to twisted (h q)-zeta function andL-functionrdquo Journal ofMathematical Analysis and Applications vol 324 no 2 pp 790ndash804 2006
[19] Y Simsek V Kurt and D Kim ldquoNew approach to the completesum of products of the twisted (h q)-Bernoulli numbers andpolynomialsrdquo Journal ofNonlinearMathematical Physics vol 14no 1 pp 44ndash56 2007
