Integral Kannappan-Sine subtraction and addition law on semigroups
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arXiv
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| Format: | Preprint |
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2025
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| author | Omar, Ajebbar Elhoucien, Elqorachi Ahmed, Jafar |
| author_facet | Omar, Ajebbar Elhoucien, Elqorachi Ahmed, Jafar |
| contents | Let $S$ be a semigroup, $μ$ a discrete measure on $S$ and $σ:S \longrightarrow S$ is an involutive automorphism. We determine the complex-valued solutions of the integral Kannappan-Sine subtraction law $$\int_{S}f(xσ(y)t)dμ(t)=f(x)g(y)-f(y)g(x),\; x,y \in S,$$ and the integral Kannappan-Sine addition law $$\int_{S}f(xσ(y)t)dμ(t)=f(x)g(y)+f(y)g(x),\; x,y \in S.$$
We express the solutions by means of exponentials on S, the solutions of the special sine addition law $f(xy)=f(x)χ(y)+f(y)χ(x),$ $x,y\in S$ and the solutions of of the special case of the integral Kannappan-Sine addition law $\int_{S}f(xσ(y)t)dμ(t)=[f(x)χ(y)+f(y)χ(x)]\int_{S}χ(t)dμ(t), $ $x,y\in S$, and where $χ$: $S\longrightarrow \mathbb{C}$ is an exponential. The continuous solutions on topological semigroups are also given. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_21057 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integral Kannappan-Sine subtraction and addition law on semigroups Omar, Ajebbar Elhoucien, Elqorachi Ahmed, Jafar General Mathematics Let $S$ be a semigroup, $μ$ a discrete measure on $S$ and $σ:S \longrightarrow S$ is an involutive automorphism. We determine the complex-valued solutions of the integral Kannappan-Sine subtraction law $$\int_{S}f(xσ(y)t)dμ(t)=f(x)g(y)-f(y)g(x),\; x,y \in S,$$ and the integral Kannappan-Sine addition law $$\int_{S}f(xσ(y)t)dμ(t)=f(x)g(y)+f(y)g(x),\; x,y \in S.$$ We express the solutions by means of exponentials on S, the solutions of the special sine addition law $f(xy)=f(x)χ(y)+f(y)χ(x),$ $x,y\in S$ and the solutions of of the special case of the integral Kannappan-Sine addition law $\int_{S}f(xσ(y)t)dμ(t)=[f(x)χ(y)+f(y)χ(x)]\int_{S}χ(t)dμ(t), $ $x,y\in S$, and where $χ$: $S\longrightarrow \mathbb{C}$ is an exponential. The continuous solutions on topological semigroups are also given. |
| title | Integral Kannappan-Sine subtraction and addition law on semigroups |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2504.21057 |