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# Prove properties of the Bernoulli polynomials

The Bernoulli polynomials are defined by 1. Find explicit formulas for the first Bernoulli polynomials in the cases .
2. Use mathematical induction to prove that is a degree polynomial in , where the degree term is .
3. For prove that .
4. For prove that 5. Prove that for .

6. Prove that for , 7. Prove that for , 1. We start with the initial condition . This gives us Now, using the integral condition to find , Thus, Next, using this expression for we have Using the integral condition to find , Thus, Next, using this expression for we have Using the integral condition to find , Thus, Next, using this expression for we have Using the integral condition to find , Thus, Finally, using this expression for we have Using the integral condition to find , Thus, 2. Proof. We have shown in part (a) that this statement is true for . Assume then that the statement is true for some positive integer , i.e., Then, by the definition of the Bernoulli polynomials we have, where for . Then, taking the integral of this expression Hence, the statement is true for the case ; hence, for all positive integers 3. Proof. From the integral property in the definition of the Bernoulli polynomials we know for , Then, using the first part of the definition we have ; therefore, Thus, we indeed have 4. Proof. The proof is by induction. For the case we have Therefore, Since , the stated difference equation holds for . Assume then that the statement holds for some positive integer . Then by the fundamental theorem of calculus, we have Therefore, Hence, the statement is true for the case , and so it is true for all positive integers 5. Proof. (Let’s assume Apostol means for to be some positive integer.) First, we use the definition of the Bernoulli polynomials to compute the integral, Now, we want to express the numerator as a telescoping sum and use part (d), Thus, we indeed have 6. Proof.

Incomplete. I’ll try to fix parts (f) and (g) soon(ish).

1. Mihajlo says:

f) can be proved by induction as well (e.g. similar to d) part).

• Mihajlo says:

Actually, while it is probably doable to do f) by induction (e.g. to prove that later odd Bernoulli numbers are 0), the other suggested solution here looks more appropriate.

• Anonymous says:

The proof of part (d) is wrong since he assumes that p_m+1(0)=0 which is clearly not true (in general) from part a

• 2. 3. 