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On Euler polynomials and their generalization. (Russian) Zbl 0767.33007
The sequence of polynomials of the type \(L_ 0\equiv 1\), \(L_{n+1}=\alpha a_ n xL_ n(x)+(a_ n x^ 2-b_ n)L_ n'(x)\), where \(\alpha>0\) is a given constant and \((a_ n)\), \((b_ n)\) is a given sequence of positive numbers, \(n=0,1,2,3,\dots\), has been studied. A special case of the polynomials are the Euler polynomials in the form by S. L. Sobolev [Sov. Math., Dokl. 18(1977), 935–938 (1978), translation from Dokl. Akad. Nauk SSSR 235, 277–280 (1977; Zbl 0398.26014)]. It is shown that at \(b_ n/a_ n\leq b_{n+1}/a_{n+1}\) the roots of all the polynomials are real and simple and the roots of neighbouring polynomials \(L_ n\) and \(L_{n+1}\) alternate, the roots of \(L_{n+1}\) being in the interval \(]-\sqrt{b_ n/a_ n},\sqrt{b_ n/a_ n}[\). Formulas are derived for the coefficients \(\ell_{np}\) of the polynomial \(L_ n\) undert certain conditions. The combinatorial connection of the \(\ell_{np}\) coefficients is discussed.
Reviewer: V.Burjan (Praha)

MSC:
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
65D20 Computation of special functions and constants, construction of tables
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