New Indefinite Summation Formulas and Some Applications
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910503762132992 |
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| author | Yadeta, Hailu Bikila |
| author_facet | Yadeta, Hailu Bikila |
| contents | In this paper, we introduce a novel indefinite summation $\sum_{t} f(t)$ (or antidifference $Δ^{-1}f(t) $ ) formula for any given function $f$. We apply the indefinite summation formula to calculate a particular solution to a nonhomogeneous linear difference equation of the form $$ y(x+h)-λy(x)=f(x),\quad h > 0,\quad λ\neq 0, $$ and also to solve a linear difference inequality of the form $$ y(x+h)-λy(x) \geq 0,\quad h > 0,\quad λ\neq 0. $$ Furthermore, we apply the formula to determine a particular solution to a difference equations of the form $$ Φ(E)y(t)=f(t), $$ and in solving a linear difference inequality of the form, $$ Φ(E)y(t)\geq 0, $$ where $ Φ(E) $ is some linear difference operator. We show how the antidifference of a function $f$ calculated with the current formula is related to the already existing result and establish the corresponding identity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18766 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New Indefinite Summation Formulas and Some Applications Yadeta, Hailu Bikila Classical Analysis and ODEs In this paper, we introduce a novel indefinite summation $\sum_{t} f(t)$ (or antidifference $Δ^{-1}f(t) $ ) formula for any given function $f$. We apply the indefinite summation formula to calculate a particular solution to a nonhomogeneous linear difference equation of the form $$ y(x+h)-λy(x)=f(x),\quad h > 0,\quad λ\neq 0, $$ and also to solve a linear difference inequality of the form $$ y(x+h)-λy(x) \geq 0,\quad h > 0,\quad λ\neq 0. $$ Furthermore, we apply the formula to determine a particular solution to a difference equations of the form $$ Φ(E)y(t)=f(t), $$ and in solving a linear difference inequality of the form, $$ Φ(E)y(t)\geq 0, $$ where $ Φ(E) $ is some linear difference operator. We show how the antidifference of a function $f$ calculated with the current formula is related to the already existing result and establish the corresponding identity. |
| title | New Indefinite Summation Formulas and Some Applications |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2406.18766 |