Whittaker Formula and the Fibonacci Numbers

Updated on March 16, 2017

In this article I want to use a specific polynomial equation to introduce the Whittaker method for finding the root that has the smallest absolute value. I will use the polynomial x2-x-1=0. This polynomial is special since the roots are x1=ϕ (golden ratio) ≈1.6180 and x2=-Φ ( negative of golden ratio conjugate)≈ - 0.6180.

Whittaker Formula

Whittaker formula is a method that uses the coefficients of the polynomial equation to create some special matrices. The determinants of these special matrices are used to create an infinite series that converges to the root that has the smallest absolute value. If we have the following general polynomial 0=a0 + a1x+ a2x2 + a3x3+a4x4 + … , the smallest root in absolute value is given by the equation found in image 1. Wherever you see a matrix in image 1, the determinant of that matrix is meant to be in its place.

The formula doesn’t work if there are more than one root with the smallest absolute value. For example, if the smallest roots are 1 and -1, you cannot use the Whittaker formula since abs(1)= abs(-1)=1. This problem can be easily bypassed by transforming the initial polynomial in another polynomial. I will deal with this problem in another article since the polynomial that I will use in this article doesn’t have this problem.

Whittaker Infinite Series Formula

Image 1
Image 1 | Source

Specific Example

The smallest root in absolute value of 0= x2-x-1 is x2=-Φ (negative of golden ratio conjugate)≈ - 0.6180. So we must obtain an infinite series that converges to x2. Using the same notation as in the previous section, we get the following assignments a0=-1, a1=-1 and a2=1. If we look at the formula from image 1 we can see that we actually need an infinite number of coefficients and we have only 3 coefficients. All the other coefficients have a value of zero, thus a3=0 , a4=0 , a5=0 etc.

The matrices from the numerator of our terms always start with the element m1,1=a2=1. In image 2 I show the determinants of the 2x2, 3x3 and 4x4 matrix that start with the element m1,1=a2=1. The determinant of these matrices is always 1 since these matrices are lower triangular matrices and the product of the elements from the main diagonal is 1n=1.

Now we should look at the matrices from the denominator of our terms. In the denominator, we always have matrices that start with the element m1,1=a1=-1. In image 3 I show the 2x2,3x3,4x4,5x5 and 6x6 matrices and their determinants. The determinants in the proper order are 2, -3, 5, -8 and 13. So we obtain successive Fibonacci numbers, but the sign alternates between positive and negative. I didn’t bother to find a proof that shows that these matrices indeed generate determinants equal to successive Fibonacci numbers (with alternating sign), but I may try in the future. In image 4 I provide the first few terms in our infinite series. In image 5 I try to generalize the infinite series using the Fibonacci numbers. If we let F1=1, F2=1 and F3=2, then the formula from image 5 should be correct.

Finally, we can use the series from image 5 to generate an infinite series for the golden number. We can use the fact that φ=Φ +1, but we also have to reverse the signs of the terms from image 5 since that is an infinite series for -Φ.

First Numerator Matrices

Image 2
Image 2 | Source

First Denominator Matrices

Image 3
Image 3 | Source

First Few Terms of The Infinite Series

Image 4
Image 4 | Source

General Formula of the Infinite Series

Image 5
Image 5 | Source

Golden Ratio Infinite Series

Image 6
Image 6 | Source

Final Remarks

If you want to learn more about the Whittaker method you should check source [1] that I provide at the bottom of this article. I think it is amazing that by using this method you can obtain a sequence of matrices that have determinants with meaningful values. Searching the internet I found the infinite series obtained in this article. This infinite series was mentioned in a forum discussion , but I could not find a more detailed article that discusses this particular infinite series.

You can try to apply this method on other polynomials and you may find other interesting infinite series. In a future article I will show how to obtain an infinite series for square root of 2 using the Pell numbers.

Questions & Answers


      0 of 8192 characters used
      Post Comment
      • RaulP profile imageAUTHOR


        3 years ago

        I discovered the Whittaker method by chance. I believe I was looking for books that covered Lill's method and I found the book listed in my sources. You can apply the Whittaker method to obtain other interesting infinite series that are made up of integers belonging to integer sequences (like Pell numbers).

      • profile image


        3 years ago

        This is nice way to derive the Fibonacci product infinite series identity. Those old out of print math books often have little hidden treasures like that.


      This website uses cookies

      As a user in the EEA, your approval is needed on a few things. To provide a better website experience, owlcation.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

      For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: https://maven.io/company/pages/privacy

      Show Details
      HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
      LoginThis is necessary to sign in to the HubPages Service.
      Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
      AkismetThis is used to detect comment spam. (Privacy Policy)
      HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
      HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
      Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
      CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
      Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
      Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
      Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
      Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
      Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
      Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
      VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
      PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
      Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
      MavenThis supports the Maven widget and search functionality. (Privacy Policy)
      Google AdSenseThis is an ad network. (Privacy Policy)
      Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
      Index ExchangeThis is an ad network. (Privacy Policy)
      SovrnThis is an ad network. (Privacy Policy)
      Facebook AdsThis is an ad network. (Privacy Policy)
      Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
      AppNexusThis is an ad network. (Privacy Policy)
      OpenxThis is an ad network. (Privacy Policy)
      Rubicon ProjectThis is an ad network. (Privacy Policy)
      TripleLiftThis is an ad network. (Privacy Policy)
      Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
      Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
      Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
      Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
      ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
      Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)
      ClickscoThis is a data management platform studying reader behavior (Privacy Policy)