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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