mathematics

Rothe–Hagen identity

Rothe–Hagen identity is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Rothe–Hagen identity rather than just read about it. In short: In mathematics, the Rothe–Hagen identity is a mathematical identity valid for all complex numbers ( x , y , z {\displaystyle x,y,z} ) except where its denominators vanish: ∑ k = 0 n x x + k z ( x + k z k ) y y + ( n − k ) z ( y + ( n − k ) z n − k ) = x + y x + y + n z ( x + y + n z n ) . {\displaystyle \sum _{k=0}^{n}{\frac {x}{x+kz}}{x+kz \choose k}{\frac {y}{y+(n-k)z}}{y+(n-k)z \choose n-k}={\frac {x+y}{x+y+nz}}{…

Key takeaways

Rothe–Hagen identity belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Rothe–Hagen identity to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Rothe–Hagen identity from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Rothe–Hagen identity is a mathematical identity valid for all complex numbers ( x , y , z {\displaystyle x,y,z} ) except where its denominators vanish:

∑ k = 0 n x x + k z ( x + k z k ) y y + ( n − k ) z ( y + ( n − k ) z n − k ) = x + y x + y + n z ( x + y + n z n ) . {\displaystyle \sum _{k=0}^{n}{\frac {x}{x+kz}}{x+kz \choose k}{\frac {y}{y+(n-k)z}}{y+(n-k)z \choose n-k}={\frac {x+y}{x+y+nz}}{x+y+nz \choose n}.}

It is a generalization of Vandermonde's identity, and is named after the independent work of Heinrich August Rothe in 1793 and of Johann Georg Hagen in 1891.

References Chu, Wenchang (2010), "Elementary proofs for convolution identities of Abel and Hagen-Rothe", Electronic Journal of Combinatorics, 17 (1), N24, doi:10.37236/473. Gould, H. W. (1956), "Some generalizations of Vandermonde's convolution", The American Mathematical Monthly, 63 (2): 84–91, doi:10.1080/00029890.1956.11988763, JSTOR 2306429, MR 0075170. See especially pp. 89–91. Hagen, Johann G. (1891), Synopsis Der Hoeheren Mathematik, Berlin, formula 17, pp. 64–68, vol. I{{citation}}: CS1 maint: location missing publisher (link). As cited by Gould (1956). Ma, Xinrong (2011), "Two matrix inversions associated with the Hagen-Rothe formula, their q-analogues and applications", Journal of Combinatorial Theory, Series A, 118 (4): 1475–1493, doi:10.1016/j.jcta.2010.12.012, MR 2763069. Rothe, Heinrich August (1793), Formulae De Serierum Reversione Demonstratio Universalis Signis Localibus Combinatorio-Analyticorum Vicariis Exhibita: Dissertatio Academica, Leipzig. As cited by Gould (1956).

Worked examples

Example 1 — a first encounter with Rothe–Hagen identity

Start with the simplest possible case. Write down what Rothe–Hagen identity claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Rothe–Hagen identity before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Rothe–Hagen identity ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Rothe–Hagen identity

In research: Rothe–Hagen identity appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Rothe–Hagen identity in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Rothe–Hagen identity is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic identities, Applied mathematics stubs, Complex analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Rothe–Hagen identity outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Rothe–Hagen identity in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rothe–Hagen identity means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Rothe–Hagen identity out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Rothe–Hagen identity in simple terms?

In mathematics, the Rothe–Hagen identity is a mathematical identity valid for all complex numbers ( x , y , z {\displaystyle x,y,z} ) except where its denominators vanish: ∑ k = 0 n x x + k z ( x + k z k ) y y + ( n − k ) z ( y + ( n − k ) z n − k ) = x + y x + y + n z ( x + y + n z n ) . {\display…

Why does Rothe–Hagen identity matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Rothe–Hagen identity?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Rothe–Hagen identity.