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Abstract

In this paper, we introduce a parametric type of Bernoulli polynomials with level 3 and study their characteristic and combinatorial properties. In particular, we show some determinant expressions of these polynomials. We also give determinant expressions of a parametric type of Bernoulli polynomials with level 2.

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Appendix

Appendix

$$\begin{aligned} B_0^{(3,0)}(p,q)&=1,\\ B_1^{(3,0)}(p,q)&=p-\frac{1}{2},\\ B_2^{(3,0)}(p,q)&=p^2-p+\frac{1}{6},\\ B_3^{(3,0)}(p,q)&=p^3-\frac{3 p^2}{2}+\frac{p}{2}-q^3,\\ B_4^{(3,0)}(p,q)&=p^4-2 p^3+p^2- 4 q^3 p+2 q^3-\frac{1}{30},\\ B_5^{(3,0)}(p,q)&=p^5-\frac{5 p^4}{2}+\frac{5 p^3}{3}-10 q^3 p^2+\left( 10 q^3-\frac{1}{6}\right) p-\frac{5 q^3}{3},\\ B_6^{(3,0)}(p,q)&=p^6-3 p^5+\frac{5 p^4}{2}-20 q^3 p^3+\left( 30 q^3-\frac{1}{2}\right) p^2-10 q^3 p+q^6+\frac{1}{42}. \end{aligned}$$
$$\begin{aligned} B_0^{(3,1)}(p,q)&=0,\\ B_1^{(3,1)}(p,q)&=q,\\ B_2^{(3,1)}(p,q)&=2q p-q,\\ B_3^{(3,1)}(p,q)&=3 q p^2-3 q p+\frac{q}{2},\\ B_4^{(3,1)}(p,q)&=4 q p^3-6 q p^2+2 q p-q^4,\\ B_5^{(3,1)}(p,q)&=5 q p^4-10 q p^3+5 q p^2-5 q^4 p+\frac{5 q^4}{2}-\frac{q}{6},\\ B_6^{(3,1)}(p,q)&=6 q p^5-15 q p^4+10 q p^3-15 q^4 p^2+q(15 q^3-1)p-\frac{5 q^4}{2}. \end{aligned}$$
$$\begin{aligned} B_0^{(3,2)}(p,q)&=0,\\ B_1^{(3,2)}(p,q)&=0,\\ B_2^{(3,2)}(p,q)&=q^2,\\ B_3^{(3,2)}(p,q)&=3 q^2 p-\frac{3 q^2}{2},\\ B_4^{(3,2)}(p,q)&=6 q^2 p^2-6 q^2 p+q^2,\\ B_5^{(3,2)}(p,q)&=10 q^2 p^3-15 q^2 p^2+5 q^2 p-q^5,\\ B_6^{(3,2)}(p,q)&=15 q^2 p^4-30 q^2 p^3+15 q^2 p^2-6 q^5 p+3 q^5-\frac{q^2}{2}\,. \end{aligned}$$
$$\begin{aligned} \widehat{B}_0^{(3,0)}(p,q)&=1,\\ \widehat{B}_1^{(3,0)}(p,q)&=p-\frac{1}{2},\\ \widehat{B}_2^{(3,0)}(p,q)&=p^2-p+\frac{1}{6},\\ \widehat{B}_3^{(3,0)}(p,q)&=p^3-\frac{3 p^2}{2}+\frac{p}{2}+q^3,\\ \widehat{B}_4^{(3,0)}(p,q)&=p^4-2 p^3+p^2+4 q^3 p-2 q^3-\frac{1}{30},\\ \widehat{B}_5^{(3,0)}(p,q)&=p^5-\frac{5 p^4}{2}+\frac{5 p^3}{3}+10 q^3 p^2-\left( 10 q^3+\frac{1}{6}\right) p+\frac{5 q^3}{3},\\ \widehat{B}_6^{(3,0)}(p,q)&=p^6-3 p^5+\frac{5 p^4}{2}+20 q^3 p^3-\left( 30 q^3+\frac{1}{2}\right) p^2+10 q^3 p+q^6+\frac{1}{42}\,. \end{aligned}$$
$$\begin{aligned} \widehat{B}_0^{(3,1)}(p,q)&=0,\\ \widehat{B}_1^{(3,1)}(p,q)&=q,\\ \widehat{B}_2^{(3,1)}(p,q)&=2q p-q,\\ \widehat{B}_3^{(3,1)}(p,q)&=3 q p^2-3 q p+\frac{q}{2},\\ \widehat{B}_4^{(3,1)}(p,q)&=4 q p^3-6 q p^2+2 q p+q^4,\\ \widehat{B}_5^{(3,1)}(p,q)&=5 q p^4-10 q p^3+5 q p^2-5 q^4 p-\frac{5 q^4}{2}-\frac{q}{6},\\ \widehat{B}_6^{(3,1)}(p,q)&=6 q p^5-15 q p^4+10 q p^3-15 q^4 p^2-q(15 q^3+1)p+\frac{5 q^4}{2}\,. \end{aligned}$$
$$\begin{aligned} \widehat{B}_0^{(3,2)}(p,q)&=0,\\ \widehat{B}_1^{(3,2)}(p,q)&=0,\\ \widehat{B}_2^{(3,2)}(p,q)&=q^2,\\ \widehat{B}_3^{(3,2)}(p,q)&=3 q^2 p-\frac{3 q^2}{2},\\ \widehat{B}_4^{(3,2)}(p,q)&=6 q^2 p^2-6 q^2 p+q^2,\\ \widehat{B}_5^{(3,2)}(p,q)&=10 q^2 p^3-15 q^2 p^2+5 q^2 p+q^5,\\ \widehat{B}_6^{(3,2)}(p,q)&=15 q^2 p^4-30 q^2 p^3+15 q^2 p^2+6 q^5 p-3 q^5-\frac{q^2}{2}\,. \end{aligned}$$

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Komatsu, T. A parametric type of Bernoulli polynomials with level 3. RACSAM 114, 151 (2020). https://doi.org/10.1007/s13398-020-00886-4

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