First of all great thanks to Luke Hoban for his ‘Twitter OAuth in F#‘. His OAuth implementation works fine for LinkedIn as well.

All you need to do is to switch the requestTokenURI/accessTokenURI/authorizeURI addresses from Twitter on LinkedIn.

Additional materials about LinkedIn public API you found at the LinkedIn developers site: https://developer.linkedin.com/

LinkedIn OAuth implementation:

open System
open System.IO
open System.Net
open System.Security.Cryptography
open System.Text

// LinkedIn OAuth Constants
let consumerKey : string = failwith "Must provide the consumerKey for an app registered at https://www.linkedin.com/secure/developer?newapp="
let consumerSecret : string = failwith "Must provide the consumerSecret for an app registered at https://www.linkedin.com/secure/developer?newapp="
let requestTokenURI = "https://api.linkedin.com/uas/oauth/requestToken"
let accessTokenURI = "https://api.linkedin.com/uas/oauth/accessToken"
let authorizeURI = "https://api.linkedin.com/uas/oauth/authorize"

// Utilities
let unreservedChars = "abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789-_.~";
let urlEncode str =
    String.init (String.length str) (fun i ->
        let symbol = str.[i]
        if unreservedChars.IndexOf(symbol) = -1 then
            "%" + String.Format("{0:X2}", int symbol)
        else
            string symbol)

// Core Algorithms
let hmacsha1 signingKey str =
    let converter = new HMACSHA1(Encoding.ASCII.GetBytes(signingKey : string))
    let inBytes = Encoding.ASCII.GetBytes(str : string)
    let outBytes = converter.ComputeHash(inBytes)
    Convert.ToBase64String(outBytes)

let compositeSigningKey consumerSecret tokenSecret =
    urlEncode(consumerSecret) + "&" + urlEncode(tokenSecret)

let baseString httpMethod baseUri queryParameters =
    httpMethod + "&" +
    urlEncode(baseUri) + "&" +
      (queryParameters
       |> Seq.sortBy (fun (k,v) -> k)
       |> Seq.map (fun (k,v) -> urlEncode(k)+"%3D"+urlEncode(v))
       |> String.concat "%26")

let createAuthorizeHeader queryParameters =
    let headerValue =
        "OAuth " +
        (queryParameters
         |> Seq.map (fun (k,v) -> urlEncode(k)+"\x3D\""+urlEncode(v)+"\"")
         |> String.concat ",")
    headerValue

let currentUnixTime() = floor (DateTime.UtcNow - DateTime(1970, 1, 1, 0, 0, 0, 0)).TotalSeconds

/// Request a token from LinkedIn and return:
/// oauth_token, oauth_token_secret, oauth_callback_confirmed
let requestToken() =
    let signingKey = compositeSigningKey consumerSecret ""

    let queryParameters =
        ["oauth_callback", "oob";
         "oauth_consumer_key", consumerKey;
         "oauth_nonce", System.Guid.NewGuid().ToString().Substring(24);
         "oauth_signature_method", "HMAC-SHA1";
         "oauth_timestamp", currentUnixTime().ToString();
         "oauth_version", "1.0"]

    let signingString = baseString "POST" requestTokenURI queryParameters
    let oauth_signature = hmacsha1 signingKey signingString

    let realQueryParameters = ("oauth_signature", oauth_signature)::queryParameters

    let req = WebRequest.Create(requestTokenURI, Method="POST")
    let headerValue = createAuthorizeHeader realQueryParameters
    req.Headers.Add(HttpRequestHeader.Authorization, headerValue)

    let resp = req.GetResponse()
    let stream = resp.GetResponseStream()
    let txt = (new StreamReader(stream)).ReadToEnd()

    let parts = txt.Split('&')
    (parts.[0].Split('=').[1],
     parts.[1].Split('=').[1],
     parts.[2].Split('=').[1] = "true")

/// Get an access token from LinkedIn and returns:
/// oauth_token, oauth_token_secret
let accessToken token tokenSecret verifier =
    let signingKey = compositeSigningKey consumerSecret tokenSecret

    let queryParameters =
        ["oauth_consumer_key", consumerKey;
         "oauth_nonce", System.Guid.NewGuid().ToString().Substring(24);
         "oauth_signature_method", "HMAC-SHA1";
         "oauth_token", token;
         "oauth_timestamp", currentUnixTime().ToString();
         "oauth_verifier", verifier;
         "oauth_version", "1.0"]

    let signingString = baseString "POST" accessTokenURI queryParameters
    let oauth_signature = hmacsha1 signingKey signingString

    let realQueryParameters = ("oauth_signature", oauth_signature)::queryParameters

    let req = WebRequest.Create(accessTokenURI, Method="POST")
    let headerValue = createAuthorizeHeader realQueryParameters
    req.Headers.Add(HttpRequestHeader.Authorization, headerValue)

    let resp = req.GetResponse()
    let stream = resp.GetResponseStream()
    let txt = (new StreamReader(stream)).ReadToEnd()

    let parts = txt.Split('&')
    (parts.[0].Split('=').[1],
     parts.[1].Split('=').[1])

/// Compute the 'Authorization' header for the given request data
let authHeaderAfterAuthenticated url httpMethod token tokenSecret queryParams =
    let signingKey = compositeSigningKey consumerSecret tokenSecret

    let queryParameters =
        ["oauth_consumer_key", consumerKey;
         "oauth_nonce", System.Guid.NewGuid().ToString().Substring(24);
         "oauth_signature_method", "HMAC-SHA1";
         "oauth_token", token;
         "oauth_timestamp", currentUnixTime().ToString();
         "oauth_version", "1.0"]

    let signingQueryParameters =
        List.append queryParameters queryParams

    let signingString = baseString httpMethod url signingQueryParameters
    let oauth_signature = hmacsha1 signingKey signingString
    let realQueryParameters = ("oauth_signature", oauth_signature)::queryParameters
    let headerValue = createAuthorizeHeader realQueryParameters
    headerValue

/// Add an Authorization header to an existing WebRequest
let addAuthHeaderForUser (webRequest : WebRequest) token tokenSecret queryParams =
    let url = webRequest.RequestUri.ToString()
    let httpMethod = webRequest.Method
    let header = authHeaderAfterAuthenticated url httpMethod token tokenSecret queryParams
    webRequest.Headers.Add(HttpRequestHeader.Authorization, header)

type System.Net.WebRequest with
    /// Add an Authorization header to the WebRequest for the provided user authorization tokens and query parameters
    member this.AddOAuthHeader(userToken, userTokenSecret, queryParams) =
        addAuthHeaderForUser this userToken userTokenSecret queryParams

let testing() =
    // Compute URL to send user to to allow our app to connect with their credentials,
    // then open the browser to have them accept
    let oauth_token'', oauth_token_secret'', oauth_callback_confirmed = requestToken()
    let url = authorizeURI + "?oauth_token=" + oauth_token''
    System.Diagnostics.Process.Start("iexplore.exe", url)

    // *******NOTE********:
    // Get the 7 digit number from the web page, pass it to the function below to get oauth_token
    // Sample result if things go okay:
    // val oauth_token_secret' : string = "9e571e13-d054-44e6-956a-415ab3ee6d23"
    // val oauth_token' : string = "044da520-0edc-4083-a061-74e115712b61"
    let oauth_token, oauth_token_secret = accessToken oauth_token'' oauth_token_secret'' ("78846")

    // Test 1: Get your profile details
    let streamSampleUrl2 = "http://api.linkedin.com/v1/people/~"
    let req = WebRequest.Create(streamSampleUrl2)
    req.AddOAuthHeader(oauth_token, oauth_token_secret, [])
    let resp = req.GetResponse()
    let strm = resp.GetResponseStream()
    let text = (new StreamReader(strm)).ReadToEnd()
    text

    // Test 2: Get a connections list
    let streamSampleUrl3 = "http://api.linkedin.com/v1/people/~/connections"
    let req = WebRequest.Create(streamSampleUrl3)
    req.AddOAuthHeader(oauth_token, oauth_token_secret, [])
    let resp = req.GetResponse()
    let strm = resp.GetResponseStream()
    let text = (new StreamReader(strm)).ReadToEnd()
    text