<p>GENERALIZATIONS OF EULER NUMBERS AND <br>POLYNOMIALS </p><p>QIU-MING LUO AND FENG QI <br>Abstract. In this paper, the concepts of Euler numbers and Euler polynomials are generalized, and some basic properties are investigated. </p><p>1. Introduction </p><p>It is well-known that the Euler numbers and polynomials can be defined by the following definitions. </p><p>Definition 1.1 ([1]). The Euler numbers E<sub style="top: 0.1245em;">k </sub>are defined by the following expansion </p><p>∞</p><p>2e<sup style="top: -0.3012em;">t </sup></p><p>e<sup style="top: -0.2398em;">2t </sup>+ 1 </p><p>E<sub style="top: 0.1246em;">k </sub></p><p>X</p><p>=</p><p>t<sup style="top: -0.3428em;">k</sup>, |t| ≤ π. </p><p>(1.1) (1.2) k! </p><p>k=0 </p><p>In [4, p. 5], the Euler numbers is defined by </p><p>ꢀ&nbsp;ꢁ </p><p>∞</p><p>n<br>2n </p><p>2e<sup style="top: -0.3013em;">t/2 </sup></p><p>e<sup style="top: -0.2398em;">t </sup>+ 1 </p><p>t</p><p>(−1) E<sub style="top: 0.1245em;">n </sub></p><p>t</p><p>X</p><p></p><ul style="display: flex;"><li style="flex:1">= sech </li><li style="flex:1">=</li></ul><p></p><p>,</p><p>|t| ≤ π. </p><p></p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">(2n)! </li><li style="flex:1">2</li></ul><p></p><p>n=0 </p><p>Definition 1.2 ([1, 4]). The Euler polynomials E<sub style="top: 0.1245em;">k</sub>(x) for x ∈ R are defined by </p><p>∞</p><p>2e<sup style="top: -0.3013em;">xt </sup></p><p>e<sup style="top: -0.2398em;">t </sup>+ 1 <br>E<sub style="top: 0.1246em;">k</sub>(x) k! </p><p>X</p><p>=</p><p>t<sup style="top: -0.3427em;">k</sup>, |t| ≤ π. </p><p>(1.3) </p><p>k=0 </p><p>It can also be shown that the polynomials E<sub style="top: 0.1245em;">i</sub>(t), i ∈ N, are uniquely determined by the following two properties </p><p></p><ul style="display: flex;"><li style="flex:1">E<sub style="top: 0.2052em;">i</sub><sup style="top: -0.3428em;">0</sup>(t) = iE<sub style="top: 0.1245em;">i−1</sub>(t), E<sub style="top: 0.1245em;">0</sub>(t) = 1; </li><li style="flex:1">(1.4) </li></ul><p></p><ul style="display: flex;"><li style="flex:1">(1.5) </li><li style="flex:1">E<sub style="top: 0.1246em;">i</sub>(t + 1) + E<sub style="top: 0.1246em;">i</sub>(t) = 2t<sup style="top: -0.3428em;">i</sup>. </li></ul><p></p><p>2000 Mathematics Subject Classification.&nbsp;11B68. </p><p>Key words and phrases.&nbsp;Euler numbers, Euler polynomials, generalization. The authors were supported in part by NNSF (#10001016) of China, SF for the Prominent </p><p>Youth of Henan Province, SF of Henan Innovation Talents at Universities, NSF of Henan Province (#004051800), Doctor Fund of Jiaozuo Institute of Technology, China. </p><p>A</p><p>This paper was typeset using A S-LT X. </p><p>M</p><p>E</p><p>1</p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">Q.-M. LUO AND F. QI </li></ul><p></p><p>Euler polynomials are related to the Bernoulli numbers. For information about <br>Bernoulli numbers and polynomials, please refer to [1, 2, 3, 4]. <br>In this paper, we will give some generalizations of the concepts of Euler numbers and Euler polynomials, and research their basic properties. </p><p>2. Generalizations of Euler numbers and polynomials </p><p>In this section, we give two definitions, the generalized Euler number and the generalized Euler polynomial, which generalize the concepts of Euler number and Euler polynomial. </p><p>Definition 2.1.&nbsp;For positive numbers a, b, c, the generalized Euler numbers E<sub style="top: 0.1245em;">k</sub>(a, b, c) are defined by </p><p>∞</p><p>2c<sup style="top: -0.3012em;">t </sup>b<sup style="top: -0.2398em;">2t </sup>+ a<sup style="top: -0.2398em;">2t </sup></p><p>E<sub style="top: 0.1246em;">k</sub>(a, b, c) </p><p>X</p><p>=</p><p>t<sup style="top: -0.3428em;">k</sup>. </p><p>(2.1) k! </p><p>k=0 </p><p>Definition 2.2. For any given positive numbers a, b, c&nbsp;and x ∈ R, the generalized Euler polynomials E<sub style="top: 0.1245em;">k</sub>(x; a, b, c) are defined by </p><p>∞</p><p>2c<sup style="top: -0.3012em;">xt </sup>b<sup style="top: -0.2398em;">t </sup>+ a<sup style="top: -0.2398em;">t </sup></p><p>E<sub style="top: 0.1246em;">k</sub>(x; a, b, c) </p><p>X</p><p>=</p><p>t<sup style="top: -0.3428em;">k</sup>. </p><p>(2.2) k! </p><p>k=0 </p><p>If taking a = 1, b = c = e, then Definition 1.1 and Definition 1.2 can be deduced from Definition 2.1 and Definition 2.2 respectively.&nbsp;Thus, the Definition 2.1 and Definition 2.2 generalize the concepts of Euler numbers and polynomials. </p><p>3. Some properties of the generalized Euler numbers </p><p>In this section, we study some basic properties of the generalized Euler numbers defined in Definition 2.1. </p><p>Theorem 3.1. For positive numbers a, b, c&nbsp;and real number x ∈ R, we have </p><p>E<sub style="top: 0.1246em;">0</sub>(a, b, c) = 1, E<sub style="top: 0.1246em;">k</sub>(1, e, e) = E<sub style="top: 0.1246em;">k</sub>, E<sub style="top: 0.1246em;">k</sub>(1, e<sup style="top: -0.3427em;">1/2</sup>, e<sup style="top: -0.3427em;">x</sup>) = E<sub style="top: 0.1246em;">k</sub>(x), </p><p>(3.1) (3.2) </p><p></p><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><p></p><p>ln c − 2 ln a <br>E<sub style="top: 0.1245em;">k</sub>(a, b, c) = 2<sup style="top: -0.3428em;">k</sup>(ln b − ln a)<sup style="top: -0.3428em;">k</sup>E<sub style="top: 0.1245em;">k </sub></p><p>,</p><p>2(ln b − ln a) </p><p>ꢀ&nbsp;ꢁ </p><p>k</p><p>X</p><p>k<br>E<sub style="top: 0.1245em;">k</sub>(a, b, c) = </p><p></p><ul style="display: flex;"><li style="flex:1">(ln b − ln a)<sup style="top: -0.3428em;">j</sup>(ln c − ln a − ln b)<sup style="top: -0.3428em;">k−j</sup>E<sub style="top: 0.1245em;">j</sub>. </li><li style="flex:1">(3.3) </li></ul><p></p><p>j</p><p>j=0 </p><p>Proof. The formulae in (3.1) follow from Definition 1.1, Definition 1.2, and Definition 2.1 easily. </p><p></p><ul style="display: flex;"><li style="flex:1">GENERALIZATIONS OF EULER NUMBERS AND POLYNOMIALS </li><li style="flex:1">3</li></ul><p></p><p>By Definition 1.2 and Definition 2.1 and direct computation, we have </p><p></p><ul style="display: flex;"><li style="flex:1">ꢂ</li><li style="flex:1">ꢃ</li></ul><p></p><p>ln c−2 ln a </p><p>2c<sup style="top: -0.3013em;">t </sup>b<sup style="top: -0.2398em;">2t </sup>− a<sup style="top: -0.2398em;">2t </sup></p><p></p><ul style="display: flex;"><li style="flex:1">2 exp </li><li style="flex:1">· 2t(ln b − ln a) </li></ul><p></p><p>2(ln b−ln a) </p><p></p><ul style="display: flex;"><li style="flex:1">ꢂ</li><li style="flex:1">ꢃ</li></ul><p></p><p>==exp 2t(ln b − ln a) +&nbsp;1 <br>(3.4) </p><p></p><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><p></p><p>∞</p><p>ln c − 2 ln a t<sup style="top: -0.3013em;">k </sup></p><p>X</p><p>2<sup style="top: -0.3427em;">k</sup>(ln b − ln a)<sup style="top: -0.3427em;">k</sup>E<sub style="top: 0.1246em;">k </sub></p><p>.</p><p>2(ln b − ln a) k! </p><p>k=0 </p><p>Then, the formula (3.2) follows. </p><p>ꢂ&nbsp;ꢃ </p><p>P</p><p>k</p><p>j=0 </p><p>kj</p><p>1</p><p>−j </p><p></p><ul style="display: flex;"><li style="flex:1">Substituting E<sub style="top: 0.1245em;">k</sub>(x) = </li><li style="flex:1">2</li><li style="flex:1">(x− <sub style="top: 0.2863em;">2 </sub>)<sup style="top: -0.3013em;">k−j</sup>E<sub style="top: 0.1245em;">j </sub>into the formula (3.2) yields the </li></ul><p>formula (3.3). </p><p>ꢀ</p><p>Theorem 3.2. For k ∈ N, we have </p><p>ꢀ&nbsp;ꢁ </p><p>k−1 </p><p>X</p><ul style="display: flex;"><li style="flex:1">ꢄ</li><li style="flex:1">ꢅ</li></ul><p></p><p>12</p><p>kj<br>E<sub style="top: 0.1245em;">k</sub>(a, b, c) = − </p><p>(2 ln b − ln c)<sup style="top: -0.3428em;">k−j </sup>+ (2 ln a − ln c)<sup style="top: -0.3428em;">k−j </sup>E<sub style="top: 0.1245em;">j</sub>(a, b, c), (3.5) </p><p>j=0 </p><p>E<sub style="top: 0.1245em;">k</sub>(a, b, c) = E<sub style="top: 0.1245em;">k</sub>(b, a, c), </p><p>(3.6) (3.7) </p><p>E<sub style="top: 0.1245em;">k</sub>(a<sup style="top: -0.3428em;">α</sup>, b<sup style="top: -0.3428em;">α</sup>, c<sup style="top: -0.3428em;">α</sup>) = α<sup style="top: -0.3428em;">k</sup>E<sub style="top: 0.1245em;">k</sub>(a, b, c). </p><p>Proof. By Definition 2.1, direct calculation yields </p><p></p><ul style="display: flex;"><li style="flex:1">"</li><li style="flex:1">#</li></ul><p></p><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">t</li><li style="flex:1">t</li></ul><p></p><p>∞</p><p>k</p><p>X</p><p>12</p><p>b<sup style="top: -0.3013em;">2 </sup></p><p>c</p><p>a<sup style="top: -0.3013em;">2 </sup></p><p>ct</p><p>1 = <br>=<br>+</p><p>E<sub style="top: 0.1245em;">k</sub>(a, b, c) </p><p>k! </p><p>k=0 </p><p></p><ul style="display: flex;"><li style="flex:1">"</li><li style="flex:1">#</li></ul><p></p><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><p></p><p>∞</p><p></p><ul style="display: flex;"><li style="flex:1">k</li><li style="flex:1">k</li></ul><p></p><p>∞</p><p></p><ul style="display: flex;"><li style="flex:1">k</li><li style="flex:1">k</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">b<sup style="top: -0.3013em;">2 </sup></li><li style="flex:1">a<sup style="top: -0.3013em;">2 </sup></li></ul><p></p><p>t</p><p></p><ul style="display: flex;"><li style="flex:1">X</li><li style="flex:1">X</li></ul><p></p><p>12</p><p>t</p><p></p><ul style="display: flex;"><li style="flex:1">ln </li><li style="flex:1">+</li></ul><p>ln ln </p><p>E<sub style="top: 0.1246em;">k</sub>(a, b, c) </p><p>(3.8) k! </p><p></p><ul style="display: flex;"><li style="flex:1">c</li><li style="flex:1">c</li></ul><p></p><p>k! </p><p></p><ul style="display: flex;"><li style="flex:1">k=0 </li><li style="flex:1">k=0 </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1"></li><li style="flex:1"></li></ul><p></p><ul style="display: flex;"><li style="flex:1">"</li><li style="flex:1">#</li></ul><p></p><p>ꢀ&nbsp;ꢁ&nbsp;ꢀ </p><p></p><ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><p></p><p>∞</p><p></p><ul style="display: flex;"><li style="flex:1">k</li><li style="flex:1">k−j </li><li style="flex:1">k−j </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">X</li><li style="flex:1">X</li></ul><p></p><p>12</p><p>k</p><p>b<sup style="top: -0.3012em;">2 </sup></p><p>c</p><p>a<sup style="top: -0.3012em;">2 </sup></p><p>t<sup style="top: -0.3012em;">k </sup></p><p></p><p></p><ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">+</li><li style="flex:1">ln </li></ul><p></p><p>E<sub style="top: 0.1246em;">j</sub>(a, b, c) <br>.</p><p></p><ul style="display: flex;"><li style="flex:1">j</li><li style="flex:1">c</li></ul><p></p><p>k! </p><p></p><ul style="display: flex;"><li style="flex:1">j=0 </li><li style="flex:1">k=0 </li></ul><p></p><p>Equating coefficients of t<sup style="top: -0.3012em;">k </sup>in (3.8) gives us </p><p></p><ul style="display: flex;"><li style="flex:1">"</li><li style="flex:1">#</li></ul><p></p><p>ꢀ&nbsp;ꢁ&nbsp;ꢀ </p><p></p><ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">k</li><li style="flex:1">k−j </li><li style="flex:1">k−j </li></ul><p></p><p>kj</p><p>b<sup style="top: -0.3013em;">2 </sup></p><p>c</p><p>a<sup style="top: -0.3013em;">2 </sup></p><p>X</p><p></p><ul style="display: flex;"><li style="flex:1">ln </li><li style="flex:1">+</li><li style="flex:1">ln </li></ul><p></p><p>E<sub style="top: 0.1246em;">j</sub>(a, b, c) = 0. </p><p>(3.9) </p><p>c</p><p>j=0 </p><p>Formula (3.5) follows. <br>The rest formulae follow from Definition 2.1 and formula (3.2). </p><p>ꢀ</p><p>Remark 3.1. For positive numbers a, b, and c, we have the following </p><p>E<sub style="top: 0.1246em;">0</sub>(a, b, c) = 1, E<sub style="top: 0.1245em;">1</sub>(a, b, c) = ln c − ln a − ln b, </p><p>(3.10) <br>E<sub style="top: 0.1245em;">2</sub>(a, b, c) = (ln c − 2 ln a)(ln c − 2 ln b), </p><p>E<sub style="top: 0.1246em;">3</sub>(a, b, c) = [(ln c − ln a − ln b)<sup style="top: -0.3427em;">2 </sup>− 3(ln b − ln a)<sup style="top: -0.3427em;">2</sup>](ln c − ln a − ln b). </p><p></p><ul style="display: flex;"><li style="flex:1">4</li><li style="flex:1">Q.-M. LUO AND F. QI </li></ul><p></p><p>4. Some properties of the generalized Euler polynomials </p><p>In this section, we will investigate properties of the generalized Euler polynomials defined by Definition 2.2. </p><p>Theorem 4.1. For any given positive numbers a, b, c&nbsp;and x ∈ R, we have </p><p>ꢀ&nbsp;ꢁ </p><p></p><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><p></p><p>kk−j </p><p>X</p><p>k</p><p></p><ul style="display: flex;"><li style="flex:1">(ln c)<sup style="top: -0.3013em;">k−j </sup></li><li style="flex:1">1</li></ul><p>2</p><p>E<sub style="top: 0.1246em;">k</sub>(x; a, b, c) = E<sub style="top: 0.1245em;">k</sub>(x; a, b, c) = E<sub style="top: 0.1246em;">k</sub>(x; a, b, c) = x − <br>E<sub style="top: 0.1246em;">j</sub>(a, b, c), </p><p>(4.1) </p><p>j</p><p>2<sup style="top: -0.2398em;">j </sup></p><p>j=0 </p><p>ꢀ&nbsp;ꢁ </p><p>ꢀ</p><p>ꢁ&nbsp;ꢀ </p><p></p><ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">k</li><li style="flex:1">j</li><li style="flex:1">k−j </li></ul><p></p><p>X</p><p></p><ul style="display: flex;"><li style="flex:1">k</li><li style="flex:1">b</li></ul><p></p><p>12ln c − 2 ln a <br>(ln c)<sup style="top: -0.3428em;">k−j </sup>ln </p><p>x − </p><p>E<sub style="top: 0.1245em;">j </sub></p><p>, (4.2) </p><p></p><ul style="display: flex;"><li style="flex:1">j</li><li style="flex:1">a</li></ul><p></p><p>2(ln b − ln a) </p><p>j=0 </p><p>ꢀ&nbsp;ꢁꢀ&nbsp;ꢁ </p><p></p><ul style="display: flex;"><li style="flex:1">ꢆ</li><li style="flex:1">ꢇ</li><li style="flex:1">ꢆ</li><li style="flex:1">ꢇ</li></ul><p></p><p>j</p><ul style="display: flex;"><li style="flex:1">k</li><li style="flex:1">`</li><li style="flex:1">k−j </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">h</li><li style="flex:1">i</li></ul><p></p><p>(ln c)<sup style="top: -0.3013em;">k−j </sup></p><p></p><ul style="display: flex;"><li style="flex:1">b</li><li style="flex:1">c</li></ul><p></p><p>12</p><p>X&nbsp;X </p><p>j−` </p><p>kjj`</p><p></p><ul style="display: flex;"><li style="flex:1">ln </li><li style="flex:1">ln </li></ul><p></p><p>x − </p><p>E<sub style="top: 0.1246em;">`</sub>, (4.3) </p><p>2<sup style="top: -0.2398em;">j </sup></p><p></p><ul style="display: flex;"><li style="flex:1">a</li><li style="flex:1">ab </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">j=0 </li><li style="flex:1">`=0 </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><p></p><p>12</p><p></p><ul style="display: flex;"><li style="flex:1">E<sub style="top: 0.1246em;">k</sub>(a, b, c) = 2<sup style="top: -0.3428em;">k</sup>E<sub style="top: 0.1246em;">k </sub></li><li style="flex:1">; a, b, c </li></ul><p>,</p><p>(4.4) (4.5) </p><p>E<sub style="top: 0.1245em;">k</sub>(x) = E<sub style="top: 0.1245em;">k</sub>(x; 1, e, e). </p><p>Proof. By Definition 2.1 and Definition 2.2, we have </p><p>∞</p><p>2c<sup style="top: -0.3013em;">2xt </sup>b<sup style="top: -0.2398em;">2t </sup>+ a<sup style="top: -0.2398em;">2t </sup>t<sup style="top: -0.3013em;">k </sup></p><p>X</p><p>=</p><p>2<sup style="top: -0.3428em;">k</sup>E<sub style="top: 0.1246em;">k</sub>(x; a, b, c) </p><p>(4.6) (4.7) k! </p><p>k=0 </p><p>and </p><p></p><ul style="display: flex;"><li style="flex:1">2c<sup style="top: -0.3013em;">2xt </sup></li><li style="flex:1">2c<sup style="top: -0.3013em;">t </sup></li></ul><p></p><p>==</p><p>· c<sup style="top: -0.3428em;">(2x−1)t </sup></p><p></p><ul style="display: flex;"><li style="flex:1">b<sup style="top: -0.2399em;">2t </sup>+ a<sup style="top: -0.2399em;">2t </sup></li><li style="flex:1">b<sup style="top: -0.2399em;">2t </sup>+ a<sup style="top: -0.2399em;">2t </sup></li></ul><p></p><p>&nbsp;</p><p>!&nbsp;&nbsp;&nbsp;</p><p>!</p><p></p><ul style="display: flex;"><li style="flex:1">∞</li><li style="flex:1">∞</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">k</li><li style="flex:1">k</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">X</li><li style="flex:1">X</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">t</li><li style="flex:1">t</li></ul><p>E<sub style="top: 0.1246em;">k</sub>(a, b, c) </p><p>(2x − 1)<sup style="top: -0.3427em;">k</sup>(ln c)<sup style="top: -0.3427em;">k </sup></p><ul style="display: flex;"><li style="flex:1">k! </li><li style="flex:1">k! </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">k=0 </li><li style="flex:1">k=0 </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1"></li><li style="flex:1"></li></ul><p></p><p>ꢀ&nbsp;ꢁ </p><p>∞</p><p>k</p><p>kj</p><p>t<sup style="top: -0.3012em;">k </sup></p><p>.</p><p></p><ul style="display: flex;"><li style="flex:1">X</li><li style="flex:1">X</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">(ln c)<sup style="top: -0.3427em;">k−j</sup>(2x − 1)<sup style="top: -0.3427em;">k−j</sup>E<sub style="top: 0.1246em;">j</sub>(a, b, c) </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1"></li><li style="flex:1"></li></ul><p></p><p>k! </p><p></p><ul style="display: flex;"><li style="flex:1">j=0 </li><li style="flex:1">k=0 </li></ul><p></p><p>t<sup style="top: -0.2505em;">k </sup></p><p>k! </p><p></p><ul style="display: flex;"><li style="flex:1">Equating the coefficients of </li><li style="flex:1">in (4.6) and (4.7) yields </li></ul><p></p><p>ꢀ&nbsp;ꢁ </p><p>k</p><p>X</p><p>k</p><ul style="display: flex;"><li style="flex:1">2<sup style="top: -0.3428em;">k</sup>E<sub style="top: 0.1245em;">k</sub>(x; a, b, c) = </li><li style="flex:1">(ln c)<sup style="top: -0.3427em;">k−j</sup>(2x − 1)<sup style="top: -0.3427em;">k−j</sup>E<sub style="top: 0.1246em;">j</sub>(a, b, c). </li></ul><p>j</p><p>j=0 </p><p>Formula (4.1) follows. <br>The rest formulae follow directly from substituting formulae (3.2) and (3.3) into </p><p>12</p><p>(4.1) and taking x = in&nbsp;(4.1), respectively. </p><p>ꢀ</p><p></p><ul style="display: flex;"><li style="flex:1">GENERALIZATIONS OF EULER NUMBERS AND POLYNOMIALS </li><li style="flex:1">5</li></ul><p></p><p>Theorem 4.2. For positive integer 1 ≤ p ≤ k, we have </p><p>p</p><p>∂</p><p>k! </p><p></p><ul style="display: flex;"><li style="flex:1">E<sub style="top: 0.1246em;">k</sub>(x; a, b, c) = </li><li style="flex:1">(ln c)<sup style="top: -0.3428em;">p</sup>E<sub style="top: 0.1246em;">k−p</sub>(x; a, b, c), </li></ul><p></p><p>(4.8) (4.9) </p><p>∂x<sup style="top: -0.2398em;">p </sup></p><p>(k − p)! </p><p>Z</p><p>x</p><p>1</p><p></p><ul style="display: flex;"><li style="flex:1">E<sub style="top: 0.1245em;">k</sub>(t; a, b, c)dt = </li><li style="flex:1">[E<sub style="top: 0.1245em;">k+1</sub>(x; a, b, c) − E<sub style="top: 0.1245em;">k+1</sub>(β; a, b, c)] . </li></ul><p></p><p>(k + 1) ln c </p><p>β</p><p>Proof. Differentiating on both sides of formula (4.1) yields </p><p>∂<br>E<sub style="top: 0.1246em;">k</sub>(x; a, b, c) = k(ln c)E<sub style="top: 0.1246em;">k−1</sub>(x; a, b, c). </p><p>(4.10) </p><p>∂x </p><p>Using formula (4.10) and by mathematical induction, the formulae (4.8) follows. Rearranging formula (4.10) produces </p><p>1</p><p>∂</p><ul style="display: flex;"><li style="flex:1">E<sub style="top: 0.1246em;">k</sub>(x; a, b, c) = </li><li style="flex:1">E<sub style="top: 0.1246em;">k+1</sub>(x; a, b, c). </li></ul><p></p><p>(4.11) <br>(k + 1) ln c ∂x </p><p>Formula (4.9) follows from integrating on both sides of formula (4.11). </p><p>ꢀ</p><p>Theorem 4.3. For positive numbers a, b, c&nbsp;and x ∈ R, we have </p><p>ꢀ&nbsp;ꢁ </p><p>k</p><p>X</p><p>k</p><ul style="display: flex;"><li style="flex:1">E<sub style="top: 0.1246em;">k</sub>(x + 1; a, b, c) = </li><li style="flex:1">(ln c)<sup style="top: -0.3427em;">k−j</sup>E<sub style="top: 0.1246em;">j</sub>(x; a, b, c), </li></ul><p></p><p>(4.12) </p><p>j</p><p>j=0 </p><p>E<sub style="top: 0.1245em;">k</sub>(x + 1; a, b, c) = 2x<sup style="top: -0.3428em;">k</sup>(ln c)<sup style="top: -0.3428em;">k </sup></p><p>ꢀ&nbsp;ꢁ </p><p>k</p><p>(4.13) (4.14) </p><p>X</p><ul style="display: flex;"><li style="flex:1">ꢄ</li><li style="flex:1">ꢅ</li></ul><p></p><p>kj</p><p></p><ul style="display: flex;"><li style="flex:1">+</li><li style="flex:1">(ln c)<sup style="top: -0.3428em;">k−j </sup>− (ln b)<sup style="top: -0.3428em;">k−j </sup>− (ln a)<sup style="top: -0.3428em;">k−j </sup>E<sub style="top: 0.1246em;">j</sub>(x; a, b, c), </li></ul><p></p><p>j=0 </p><p></p><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><p></p><p>a b <br>E<sub style="top: 0.1245em;">k</sub>(x + 1; a, b, c) = E<sub style="top: 0.1245em;">k </sub>x; , ,&nbsp;c <br>.</p><p>c c </p><p>Proof. From Definition 2.2 and straightforward calculation, we have </p><p>"</p><p>#&nbsp;" </p><p>#</p><p></p><ul style="display: flex;"><li style="flex:1">∞</li><li style="flex:1">∞</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">k</li><li style="flex:1">k</li></ul><p></p><p>2c<sup style="top: -0.3012em;">xt </sup>b<sup style="top: -0.2398em;">t </sup>+ a<sup style="top: -0.2398em;">t </sup></p><p></p><ul style="display: flex;"><li style="flex:1">t</li><li style="flex:1">t</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">X</li><li style="flex:1">X</li></ul><p></p><p>· c<sup style="top: -0.3428em;">t </sup>= </p><p>E<sub style="top: 0.1245em;">k</sub>(x; a, b, c) </p><p>(ln c)<sup style="top: -0.3428em;">k </sup></p><ul style="display: flex;"><li style="flex:1">k! </li><li style="flex:1">k! </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">k=0 </li><li style="flex:1">k=0 </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1"></li><li style="flex:1"></li></ul><p></p><p>(4.15) </p><p>ꢀ&nbsp;ꢁ </p><p>∞</p><p>k</p><p>X&nbsp;X </p><p>kj</p><p>t<sup style="top: -0.3013em;">k </sup></p><p></p><ul style="display: flex;"><li style="flex:1"></li><li style="flex:1"></li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">(ln c)<sup style="top: -0.3428em;">k−j</sup>E<sub style="top: 0.1245em;">j</sub>(x; a, b, c) </li></ul><p>k! </p><p></p><ul style="display: flex;"><li style="flex:1">j=0 </li><li style="flex:1">k=0 </li></ul><p></p><p>and </p><p>∞</p><p>k</p><p>2c<sup style="top: -0.3012em;">xt </sup></p><p>2c<sup style="top: -0.3012em;">(x+1)t </sup></p><p>t</p><p>X</p><p>· c<sup style="top: -0.3428em;">t </sup>= </p><p>=</p><p>E<sub style="top: 0.1245em;">k</sub>(x + 1; a, b, c). </p><p>(4.16) </p><p></p><ul style="display: flex;"><li style="flex:1">b<sup style="top: -0.2398em;">t </sup>+ a<sup style="top: -0.2398em;">t </sup></li><li style="flex:1">b<sup style="top: -0.2398em;">t </sup>+ a<sup style="top: -0.2398em;">t </sup></li></ul><p></p><p>k! </p><p>k=0 </p><p>t<sup style="top: -0.2506em;">k </sup></p><p>k! </p><p>Therefore, from equating the coefficients of follows. in (4.15) and (4.16), formula (4.12) </p><p></p><ul style="display: flex;"><li style="flex:1">6</li><li style="flex:1">Q.-M. LUO AND F. QI </li></ul><p></p><p>Similarly, we obtain </p><p>∞</p><p>k</p><p>2c<sup style="top: -0.3013em;">(x+1)t </sup></p><p>t</p><p>2c<sup style="top: -0.3013em;">xt </sup></p><p>X</p><p>=</p><p>E<sub style="top: 0.1245em;">k</sub>(x + 1; a, b, c) = 2c<sup style="top: -0.3428em;">xt </sup></p><p>+</p><p>(c<sup style="top: -0.3428em;">t </sup>− b<sup style="top: -0.3428em;">t </sup>− a<sup style="top: -0.3428em;">t</sup>) b<sup style="top: -0.2398em;">t </sup>+ a<sup style="top: -0.2398em;">t </sup></p>