Polynomial Functions

Functions > Special Functions > Other Special Functions > Polynomial Functions

Polynomial Functions

The following functions return the value of the listed polynomial of degree n at x:

• Her(n, x)—Hermite polynomial, or the solution of the following equation:

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  <math resultRef="11">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <plus />
        <apply>
          <minus />
          <apply>
            <derivative />
            <lambda>
              <boundVars>
                <id xml:space="preserve">x</id>
              </boundVars>
              <id xml:space="preserve">y</id>
            </lambda>
            <degree>
              <real>2</real>
            </degree>
          </apply>
          <apply>
            <mult />
            <apply>
              <scale />
              <real>2</real>
              <id xml:space="preserve">x</id>
            </apply>
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <id xml:space="preserve">y</id>
              </lambda>
              <degree>
                <placeholder />
              </degree>
            </apply>
          </apply>
        </apply>
        <apply>
          <scale />
          <real>2</real>
          <id xml:space="preserve">ny</id>
        </apply>
      </apply>
      <real>0</real>
    </apply>
  </math>
</region>

• Lag(n, x)—Laguerre polynomial, or the solution to the following equation:

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  <math resultRef="13">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <plus />
        <apply>
          <plus />
          <apply>
            <mult />
            <id xml:space="preserve">x</id>
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <id xml:space="preserve">y</id>
              </lambda>
              <degree>
                <real>2</real>
              </degree>
            </apply>
          </apply>
          <apply>
            <mult />
            <parens>
              <apply>
                <minus />
                <real>1</real>
                <id xml:space="preserve">x</id>
              </apply>
            </parens>
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <id xml:space="preserve">y</id>
              </lambda>
              <degree>
                <placeholder />
              </degree>
            </apply>
          </apply>
        </apply>
        <id xml:space="preserve">ny</id>
      </apply>
      <real>0</real>
    </apply>
  </math>
</region>

• Leg(n, x)—Legendre polynomial, or the solution to the following equation:

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  <math resultRef="15">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <plus />
        <apply>
          <minus />
          <apply>
            <mult />
            <parens>
              <apply>
                <minus />
                <real>1</real>
                <apply>
                  <pow />
                  <id xml:space="preserve">x</id>
                  <real>2</real>
                </apply>
              </apply>
            </parens>
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <id xml:space="preserve">y</id>
              </lambda>
              <degree>
                <real>2</real>
              </degree>
            </apply>
          </apply>
          <apply>
            <mult />
            <apply>
              <scale />
              <real>2</real>
              <id xml:space="preserve">x</id>
            </apply>
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <id xml:space="preserve">y</id>
              </lambda>
              <degree>
                <placeholder />
              </degree>
            </apply>
          </apply>
        </apply>
        <apply>
          <scale />
          <apply>
            <id xml:space="preserve">n</id>
            <apply>
              <plus />
              <id xml:space="preserve">n</id>
              <real>1</real>
            </apply>
          </apply>
          <id xml:space="preserve">y</id>
        </apply>
      </apply>
      <real>0</real>
    </apply>
  </math>
</region>

• Tcheb(n, x)—Chebyshev of the first kind, or the solution to the following equation:

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  <math resultRef="17">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <plus />
        <apply>
          <minus />
          <apply>
            <scale />
            <parens>
              <apply>
                <minus />
                <real>1</real>
                <apply>
                  <pow />
                  <id xml:space="preserve">x</id>
                  <real>2</real>
                </apply>
              </apply>
            </parens>
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <id xml:space="preserve">y</id>
              </lambda>
              <degree>
                <real>2</real>
              </degree>
            </apply>
          </apply>
          <apply>
            <mult />
            <id xml:space="preserve">x</id>
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <id xml:space="preserve">y</id>
              </lambda>
              <degree>
                <placeholder />
              </degree>
            </apply>
          </apply>
        </apply>
        <apply>
          <scale />
          <apply>
            <pow />
            <id xml:space="preserve">n</id>
            <real>2</real>
          </apply>
          <id xml:space="preserve">y</id>
        </apply>
      </apply>
      <real>0</real>
    </apply>
  </math>
</region>

• Ucheb(n, x)—Chebyshev of the second kind, solution to the following equation:

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  <math resultRef="19">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <plus />
        <apply>
          <minus />
          <apply>
            <scale />
            <parens>
              <apply>
                <minus />
                <real>1</real>
                <apply>
                  <pow />
                  <id xml:space="preserve">x</id>
                  <real>2</real>
                </apply>
              </apply>
            </parens>
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <id xml:space="preserve">y</id>
              </lambda>
              <degree>
                <real>2</real>
              </degree>
            </apply>
          </apply>
          <apply>
            <mult />
            <apply>
              <scale />
              <real>3</real>
              <id xml:space="preserve">x</id>
            </apply>
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <id xml:space="preserve">y</id>
              </lambda>
              <degree>
                <placeholder />
              </degree>
            </apply>
          </apply>
        </apply>
        <apply>
          <scale />
          <apply>
            <id xml:space="preserve">n</id>
            <apply>
              <plus />
              <id xml:space="preserve">n</id>
              <real>2</real>
            </apply>
          </apply>
          <id xml:space="preserve">y</id>
        </apply>
      </apply>
      <real>0</real>
    </apply>
  </math>
</region>

• Jac(n, a, b, x)—Jacobi with parameters a and b, solution to the following equation:

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  <math resultRef="21">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <plus />
        <apply>
          <plus />
          <apply>
            <scale />
            <parens>
              <apply>
                <minus />
                <real>1</real>
                <apply>
                  <pow />
                  <id xml:space="preserve">x</id>
                  <real>2</real>
                </apply>
              </apply>
            </parens>
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <id xml:space="preserve">y</id>
              </lambda>
              <degree>
                <real>2</real>
              </degree>
            </apply>
          </apply>
          <apply>
            <scale />
            <parens>
              <apply>
                <minus />
                <apply>
                  <minus />
                  <id xml:space="preserve">b</id>
                  <id xml:space="preserve">a</id>
                </apply>
                <apply>
                  <scale />
                  <parens>
                    <apply>
                      <plus />
                      <apply>
                        <plus />
                        <id xml:space="preserve">a</id>
                        <id xml:space="preserve">b</id>
                      </apply>
                      <real>2</real>
                    </apply>
                  </parens>
                  <id xml:space="preserve">x</id>
                </apply>
              </apply>
            </parens>
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <id xml:space="preserve">y</id>
              </lambda>
              <degree>
                <placeholder />
              </degree>
            </apply>
          </apply>
        </apply>
        <apply>
          <scale />
          <apply>
            <id xml:space="preserve">n</id>
            <apply>
              <plus />
              <apply>
                <plus />
                <apply>
                  <plus />
                  <id xml:space="preserve">n</id>
                  <id xml:space="preserve">a</id>
                </apply>
                <id xml:space="preserve">b</id>
              </apply>
              <real>1</real>
            </apply>
          </apply>
          <id xml:space="preserve">y</id>
        </apply>
      </apply>
      <real>0</real>
    </apply>
  </math>
</region>

Chebyshev and Legendre polynomials are special cases of the Jacobi polynomials.

Arguments

• n is a nonnegative integer.

• x is a real scalar.

• a, b are real scalars greater than –1.

Related Topics

About Other Special Functions

Example: Polynomial Functions

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