# THE ZEROS OF CERTAIN FAMILY OF SELF-RECIPROCAL POLYNOMIALS

• Published : 2007.08.31
• 74 10

#### Abstract

For integral self-reciprocal polynomials P(z) and Q(z) with all zeros lying on the unit circle, does there exist integral self-reciprocal polynomial $G_r(z)$ depending on r such that for any r, $0{\leq}r{\leq}1$, all zeros of $G_r(z)$ lie on the unit circle and $G_0(z)$ = P(z), $G_1(z)$ = Q(z)? We study this question by providing examples. An example answers some interesting questions. Another example relates to the study of convex combination of two polynomials. From this example, we deduce the study of the sum of certain two products of finite geometric series.

#### Keywords

self-reciprocal polynomials;convex combination;zeros;unit circle

#### References

1. A. Cohn, U ber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise, Math. Z. 14 (1922), no. 1, 110-148 https://doi.org/10.1007/BF01215894
2. S.-H. Kim, Factorization of sums of polynomials, Acta Appl. Math. 73 (2002), no. 3, 275-284 https://doi.org/10.1023/A:1019770930906
3. P. Lakatos, On zeros of reciprocal polynomials, Publ. Math. Debrecen 61 (2002), no. 3-4, 645-661
4. M. Marden, Geometry of Polynomials, Math. Surveys, No. 3, Amer. Math. Society, Providence, R.I., 1966
5. H. J. Fell, On the zeros of convex combinations of polynomials, Pacific J. Math. 89 (1980), no. 1, 43-50 https://doi.org/10.2140/pjm.1980.89.43

#### Cited by

1. ON SELF-RECIPROCAL POLYNOMIALS AT A POINT ON THE UNIT CIRCLE vol.46, pp.6, 2009, https://doi.org/10.4134/BKMS.2009.46.6.1153
2. ON SOME COMBINATIONS OF SELF-RECIPROCAL POLYNOMIALS vol.27, pp.1, 2012, https://doi.org/10.4134/CKMS.2012.27.1.175
3. ON THE ZEROS OF SELF-RECIPROCAL POLYNOMIALS SATISFYING CERTAIN COEFFICIENT CONDITIONS vol.47, pp.6, 2010, https://doi.org/10.4134/BKMS.2010.47.6.1189